2013-03-03

The Life of Pi

Beware the ides of March and a Happy Saint Patrick’s Day t'ya, but as winter snows abate and the vernal equinox comes nigh, a young mathematics teacher’s fancy turns to thoughts of...π.

Spreading like the ``Harlem Shake’’ meme, it seems every classroom across the land now observes an in-school holiday called Pi Day.  For the uninitiated, Pi Day is cleverly celebrated on March 14, or 3/14.  A quick trip over to Pinterest regales you with endless examples of the hackneyed puns, cartoons, song lyrics, decimal expansions, and other non-activities that grace school walls to mark this annular (HA!) event.

Never mind that in Europe and many other parts of the world, Pi Day would be celebrated on 31/4 or the 31st of April, or perhaps on 3/14, the 3rd day of Dodecember, 14th month of the year—if such dates actually occurred—ergo maybe π only exists within US borders.  Let's not start World War π over such differences.


The notion of Pi Day offends our mathematics sensibilities on many levels, so if it were up to us, this half-baked practice would be shelved, and replaced with more intellectually circuitous activities.  (Also here and here.  And newly added, here.)  We don’t want to be party poopers or sticks-in-the-mud, though, so we’ll file away our petition, but proceed with our reasoning anyway.

Unbeknownst to slightly more than three people, π is a ratio that exists in nature; if you traveled to another planet inhabited by 5-armed 3-eyed roly-poly beings, their mathematicians would know π, too.  The dissonance arises in decimal approximations for π, whether it be 3.14, 3.14159 (Yay, Purdue!), or π calculated to lots and lots of digits.  These approximations manifest one characteristic that is definitely not universal, and that is they are represented in base 10, which is just one base among infinitely many, but happens to be the number system most earthlings use.

If you write π in base 9, the first 3 digits are 3.12; in base 11, 3.16.  In base 2, π begins 11.001001000011...; in hexadecimal, it begins 3.243F6A8885A300... (yes, those are letters in there).  Are there bases in which Pi Day could be celebrated on the same day in the US and France?  Coincidentally, π in base e (the e of natural logarithms, or the ``other’’ commonly studied number which exists in nature) begins 10.10..., so that would make "Pi-e Day" October 10 anywhere.  (π in base π = 10.)  But even π in base e wouldn’t resolve the unspoken contradiction of Pi Day: π is a universal constant, but any rendering of π as a decimal approximation is arbitrary.  If students should learn one important dichotomy in mathematics, it is to understand the difference between the immutable and the intractable (a notion that can certainly be extended beyond the mathematical sphere).

Mathematics is rife with arbitrary choices.  After learning base 10 place value and arithmetic, one of the next arbitrary numbers students encounter is 360, the number of degrees in a circle—but how many teachers stop to explain its history?  Martians might have divided a circle into 680 degrees.  Such choices may have odd consequences: a triangle’s angles add to 180, a right angle has 90 degrees, your shadow is as long as your height only when the sun is at a 45 degree angle—the arbitrary choice, um, casts its shadow far and wide.  Not every student makes it as far as trigonometry, often where radian measure is introduced, which finally resolves this particular discordance, thanks to π.

***

Readers, we hope you realize much of our critique of Pi Day has been somewhat tongue-in-cheek, but our sense is that many Pi Day observances emphasize form over substance.  However, you shouldn’t think we are dissing π or want students to be kept out of the inner circle of knowledge; in fact, it’s quite the opposite.  We want π to be afforded the proper respect and honor it so roundly deserves.  The challenge lies in the delivery: if students don't gain a level of discernment early on, they may never fully understand not only the hows and whys of π, but of 60 minutes in an hour, 12 inches in a foot, why there was a leap year in 2000 and generally every four years but there won’t be a February 29 in 2100, and won't acquire many of the myriad elements of number sense that are the hallmark of a socially cognizant and mathematically proficient student.

That's why we are particularly incensed, first by Common Core’s general neglect, and then by its shabby treatment of such an essential topic: the concept of π.

First, our approach: how would we at ccssimath.blogspot.com like π to first enter students' consciousness?  Instead of Pi Day posters and platitudes, one of the simplest but most effective activities would be to let students calculate π's first few digits themselves—without our spoiling the story's ending (suggested lesson plan to follow).

At what age is it appropriate for π to be introduced, in terms of both conceptual understanding and performing calculations?  We’re not here today to talk about Common Core’s treatment of decimal arithmetic, but discovering π is a classroom activity that can follow soon after the learning of decimal division, which Common Core covers in 5.NBT.7.  We agree that Grade 5 is the right year to learn how to divide decimals, but rather than covering this obtuse procedure solely as a rote exercise, calculating π is a worthwhile and memorable application of decimal division.

After a class discussion of circle nomenclature, our suggested classroom activity for Grade 5 introduces π surreptitiously:
With centimeter tape measures in hand, groups of students carefully measure the circumferences and diameters of various round objects.  The class makes a table of measured values and calculates the quotientsWhen they see time and time again the same answer result from division, whether it be from big circular objects or small ones—eureka!—they will have unwittingly discovered π for themselves.
We think students won't likely forget where π comes from when they've taken a central role in the discovery process, as opposed to aimless Pi Day references to the symbol, to pictures of circles, or to π's digits, perhaps before they understand the meaning of a decimal point.

Purists may counter: what about a real derivation of π or π's irrationality?  Nothing says those deeper investigations can't be revisited, but we're talking fifth grade here, folks.

Calculating π in class is an age-appropriate activity, but to reach a useful mathematical understanding, several skills need to coalesce and be applied: decimal division and ratios.  Measurement is a skill we've assumed students can already do with some accuracy, and we hope they can make tables of values.  Actually, we have it a bit backward: ratios are a concept that don’t need to be previously known; but calculating π is the perfect activity to introduce ratios.

An added bonus of introducing ratios while learning about π is that it skips past the simplistic notion that ratios should have integral values.  When students see whole number ratios first, they may be lulled into the false notion that ratios are always in whole denominations.  That, in fact, is the path on which CCSSI would have students walk: 6.RP.1 introduces the concept of ratios, and the first example Common Core offers is “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.”  Another dumbing down of the standards.

Common Core, in contrast to our plan for Grade 5, doesn’t introduce ratios until Grade 6, which is too late and too simplistic.  Worse still, Common Core doesn’t introduce π until—not fifth, not sixth—but Grade 7.  It’s far too late.

Decimal division, ratios and π—they belong together.

•••

In contrast to our preference, π is rarely introduced as a ratio in American math education.  How might we substantiate this allegation?  Teachers, we’d like you to help us with a survey.  At the beginning of a class, without warning, tell your students to take out a piece of paper and in 20 seconds write their answers to this orally posed question: ``What is π?’’  No hints, no rules, no suggestions; they’re not being scored.  How many will write the Greek letter or the transliteration ``pi’’?  How many will write a decimal approximation?  How many will write 22/7 or some other fractional approximation?  Tabulate the results, and write them and the grade level in the comments section at the end of this post.

The real distinction we want to draw is: How many students wrote any of the answers above versus how many wrote an answer like ``π is the ratio of a circle’s circumference to its diameter’’.  We’d expect that not many wrote this last type of answer, because it’s not the first thing to come to mind.  We can't prove it, though, so that’s the question we’d like answered.

The point we make is that it will lead to an indelible understanding for students to first learn the meaning of π, not its representation.

Incidentally, students in China (and some other countries) may have an unintentional advantage.  The mathematical term that elementary school students first learn when calculating and discovering the concept of π is a three character word that literally means ``circle circumference ratio’’.  They use the Chinese term long before using the Greek letter π, so in effect, they’re imprinting ``circle circumference ratio’’ as π's definition.

***

If not as a ratio, how does π formally enter many American math classes?  Primarily in terms of two formulas that are written, memorized and practiced together, then provided on exam cheat sheets anyway and tested.  Despite of all the preparation, questions with π in them are too often bungled.

In the following open-ended question asking for the circumference of a circle (the diameter was 5 cm), the NAEP analysis provided four representative student answers, and three of those answers cited the circle area formula.  None of the four cited the correct circumference formula, and overall, only 28% of 12th graders got the right answer, considered to be in the range from 15.0 to 16.4 cm (allowing for measurement error).


That 72% of graduating American high school seniors cannot calculate a circle's circumference renders Pi Day "celebrations" somewhat premature.

Lesson unlearned, this faulty approach to ``teaching'' π is exactly what Common Core advocates.  π (implicitly) premieres in standard 7.G.4, which states, ``Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.’’  Common Core wants students to ``know’’ (read: memorize) the circumference and area formulas and be able to plug numbers into them, without ever needing to understand what π really means.

(Under Common Core, Grade 7 students may not have studied circles for three years: no mention of circles has been made since the fourth grade’s 4.MD.5a.)

As for the second part of 7.G.4, the wording is somewhat misleading: students shouldn’t think of the relationship between circumference and area because circumference is a length, and one does not use the circumference to determine the area; but both the circumference and area of a circle are related to π.  Perhaps the standard implies the area formula may be derived from the circumference formula (which itself could have been determined in class after the Grade 5 activity previously described), but that’s a teacher-led exercise to be done in class (also preferably in Grade 5, see below), and not on a test.

One informal derivation of the circle area formula comes from cutting a circle into successively thinner equal wedges, rearranging them into the approximate shape of a parallelogram-cum-rectangle, and applying the area formula b x h:

Completing the proof is left to the reader.  Another bonus: Grade 5 students aren’t formally learning limits or calculus yet, but in terms of concept, this derivation of the circle area formula is awfully close.

Go ahead, celebrate Pi Day, as even the US Department of Education suggests (we'll pass).  But let’s show students what π really means and how it's really used, instead of following Common Core’s plan to continue throwing π’s at them and making us eat humble pie when they don't get it.

2 comments:

  1. So you don't celebrate square root of two day, either, on Jan. 24 (24/1) ? You just won't let people force rationality on you...

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  2. kay kirkpatrick06 April, 2013

    Know I'm a bit late with this comment, but there's always next year:)Thanks for not ignoring the rest of the world like most pi day discussions do! I love the suggestions of 31st April and 3rd of Dodecember. Thanks also for the information about the Chinese characters - didn't know that. Great to talk about with our Chinese background students too. I agree wholeheartedly with your concerns, so hope you won't think I've missed the point, but there is something that is definitely worth a pi fight over and that is "pi approximation day"! In Australia we always celebrated 'pi day' on the 22nd of July (22/7 for us) but apparently the US legal declaration of 3/14 as 'national pi day' has somehow managed to make it 'international pi day', with a passing reference on some US sites to 'pi approximation day' for 22/7. Surely this implies that 3.14 isn't an approximation!! Especially when 22/7 closer!
    I live with the hope that pi day discussions could at least look at a comparison of the two. It's certainly more likely to happen if your suggestions are followed. Students could compare their measured and calculated table of results to both.

    (By the way, how many teachers do you think might allow their students to calculate the decimal value of their ratio with a calculator, without any discussion of the rounding or truncating that is involved with that too! Another reason I like 22/7 - sevenths take a reasonable amount of calculation before they start to repeat.)

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