If the red dots represent housing prices in a neighborhood, what is the median price?  Interestingly enough, we can figure this out without knowing what the scale is in the representation.  All we need to do is point at a particular red dot, a particular house, and say “this one’s price is the median price.”  And how do we find this median house?  We can use the two-handed approach, but this time hopping from dot to dot rather than from blue line to blue line.  We start with our left hand on the dot on the extreme left, and with our right hand on the dot on the extreme right, moving one dot at a time towards the center, in unison with both hands.  This center dot is the one where an equal number of dots are to the left as to the right.  It’s in the center in that particular meaning of the word.

If the data set is large enough, we can talk about median in terms of “half”: if your household income is exactly the median household  income in the country, then half of the households in the country make less, and half of the  households in the country make more.  Or, another way of looking at the same phenomenon, if I visited all households in the country, sorted from poorest to richest, by the time I was halfway done, I would be visiting the household with the median income.  (I better do all this traveling very fast, otherwise the median income will surely have changed during my trip.)

For smaller data sets, the notion of center can be a bit rough.  The one shown above can show us some of the effects through comparison with the one following:

The two data sets have an identical median.  This, even though in the new data set all the high values are really high, and all the low values are only moderately low.  For, the same data point is still the one with four values lower and four values higher.  Intuitively, it may feel that the center must have moved up compared to the earlier one.  Well, depending on the situation, the idea of what would make a good center may not be that clear-cut.

Another “gotcha” comes when we look at a data set with an even number of data points, as this one:

Note that one of the high-end data points is missing; and this time, as we do our two-handed travel from the end points towards the center, we end up with two red dots rather than one.  The standard way that this case is treated is by looking for the center between the two remaining red dots.  This is indicated in the picture by the black arrow.  If the data set represents housing prices, this time the median housing price is not the price of a particular house that we could point to – this time the median price is an in-between price.  Note that the median can still be found with our two-handed approach, but once we get to the center two dots, we must switch from hopping from dot to dot to hopping from blue delineation to blue delineation.  Simple enough, but in a sense very strange – probably the only place in the K-12 curriculum where we summarily shift from one kind of a unit (here, houses in a data set) to an entirely different kind of unit (here, dollars or thousands of dollars) without much fanfare.  At least it is only an artifact of the data set being small and discrete – if there are enough data points, you would expect the middle value, the value just below and the value just above to all be smack on top of each other so that it wouldn’t matter particularly which one you picked.  Off by half a house?  Not a big deal if there are enough of them to point to.

Unlike other things you learn in math class (like the mode of a data set), the median is widely useful in the world outside of school.  Interesting that this single notion of a two-handed approach for finding the center, which students understand so deeply, was so useful here.

Can you name the background assumptions this comic strip makes, things that do not need to be stated explicitly for a wide audience to appreciate the comic?

I’d start with the assumption of math homework.  Then, that the child needs help with the math homework.  That the parents are the designated math homework helpers.  That to be of help with the math homework, you need to know and understand the math yourself.  However, parents either did or did not survive the math gauntlet at the level of math that their kid now faces.  And that for folks who didn’t survive the math gauntlet, a fear of math is at least as real as fear of spiders, and at least as permanent a condition.

For each of these underlying assumptions, something interesting could be written and at non-trivial length.  I may be inspired to do so on some of these myself.  But don’t worry, I won’t do so here.  Let me just say that this entire blog is dedicated to the notion that whatever bad reaction you have had to mathematics in school does not need to be a permanent condition.  “Math as a garden, friendly and always new” is this blog’s motto and its commitment.

Let’s suppose I have a folder of test sheets from a standardized test.  Each piece of paper in the folder is from a particular student, and is marked with a score from 0-20.  On a long table, I make room for 21 stacks of test sheets, side by side, by putting down yellow stickies each marked with one of the possible scores from low to high.  I then take the test sheets, one by one, and place them in the stack corresponding with the score on the sheet.  If Jesse’s sheet is marked with score 15, that sheet will go on the stack labeled 15 on the yellow sticky.  When I’m done placing all the test sheets, I end up with a number of stacks of varying heights – and some stacks may be empty.  Below is a representation for what I ended up with:

The usual name for this representation is histogram.  My highest stack had eight test sheets in it, for score 15.  Score 10′s stack was empty.  Jesse’s sheet is somewhere in the 15 stack, but my representation doesn’t show where in the stack it is.  Each box in my representation could have been marked with the student’s name, but I didn’t do that.  As with any representation, this representation highlights certain information and leaves out other information altogether.  One thing left out is the name of the student, another thing is any information on the order in which the sheets were in the original folder.  If I had shuffled the sheets in the folder, the representation as shown would have been identical.

What information can we extract from these representations?  Quite a bit, actually.  Some are simple and some are useful.  A simple thing we can see from the representation is that nobody had a score of 10, nor a score of 11.  Of course, neither the histogram representation, nor the stacks of sheets with the yellow sticky was critical to finding this out.  I could have answered the question “how many students have a score of 10?” by flipping through the entire stack of sheets and counting the ones that have a score of 10.  If that was the sole thing I cared about, it might even have been a bit faster and simpler doing it that way.  Yet, after I answered the question “how many 10′s?” I would have had to do the same amount of work all over again if there was a follow-up question; “how many students have a score of 20?”

One side effect of putting the test sheets in stacks based on their score is that we have now effectively sorted them by score.  If we were to put them all back in a single stack again, by collecting the individual stacks in score-order, the resulting stack is sorted by score.  If I divide the sorted stack in half, I’ve located the middle value (called median – though median is a technical term, and I have left out the precise rules for what to do when the number of sheets in the stack is even, and what to do if it’s odd.)  If I divide the sorted stack in five equal parts, I get the quintiles, if I divide the sorted stack in four equal parts, I get the quartiles.

The histogram also allows me to answer questions like “what percentage of students scored 16 or above?”  All of those questions could have been answered from the original pile of test sheets, but the histogram makes it more straightforward.

The process of getting the histogram has clear parallels with the various sieves I’ve shown in earlier posts.  Enough so, that my claim that sieves and look-up tables don’t have much support in the K-12 curriculum was too hasty.

In the USA we do this with presidential candidates: we make them campaign on the cold streets of Iowa, we make them raise money and watch how effectively they spend it. The 2008 presidential election was effectively a two-year marathon.
Any tournament has similar characteristics: by the time you get to the finals, you should expect a stronger opponent than the one you beat in the semi-finals.

Outside of school, but in the shadow of school, we have other activities that have selection built in, that are organized as a series of ever-higher hurdles. I’m thinking of ballet, martial arts, piano, guitar, swimming, basket ball, but also of World of Warcraft and StepMania. All or most of these involve large numbers of enthusiastic kids who start at the lower levels, and far fewer kids who stick it out through the higher levels.
These selection systems work, and kids sort themselves out. Jeremy plays Bach on the violin but can’t do the splits to save his life, and Joanie is on her way to be an Olympic-level swimmer, but dropped out of Taekwondo after a single lackluster year.

While kids pursue these activities and attempt the ever-higher hurdles, they aren’t necessarily left alone, nor thrown willy-nilly into the deep end. They get encouraged, challenged, coached, cajoled, applauded. Somebody cares about their performance, somebody drives them to soccer practice, somebody buys them their uniform.

You could structure an educational system on these same principles, and I would argue that in many ways we have. Historically, we thought it was perfectly appropriate for school to act as a sorting mechanism in this way. After a few years of school, we could tell that Johnny should become a carpenter, that Jessica should become a nurse, that Jason should pursue something with numbers, and that Josh had a knack for words.

We’ve changed our thinking about what education is and what school should be for. We now talk in terms of a minimum set of knowledge, skills and capacities that students should be taught before we let them loose. They should acquire all this while in school from K-12, and we should leave no child behind while doing this.

What we seem to miss in all of this is that the infrastructure of school, the institution of school, is still mostly geared around the old model of selection. School is still better at sorting than at educating. And mathematics, for better or worse, is in the center of this.