Republic of Mathematics blog

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On January 29, Michel Paul sent me the following email:

“The first posting I saw of yours was No tests, quizzes, exams and no homework about a year ago. It deeply resonated with me, as in trying to incorporate GeoGebra, Python, and Sage in my classes, the idea of more project-based assessment started to occur.

When receiving projects using something like GeoGebra I have been pleased by the creativity I’ve been getting back. Not in all cases, of course, but in many. Projects give me a completely different perspective of what’s going on in students’ minds than test results do.

I’ve come to the conclusion that when we base mathematical assessment on tests that measure how many correct / how many possible, we’re not really measuring mathematical thinking. We’re just measuring a kind of schoolish competence. Unfortunately, this kind of assessment is somehow perceived as ‘objective’ and is the deeply entrenched norm. Suggesting the use of project based assessment invites ridicule and complaints of a ‘lowering of standards’. However, I see lots of cases of kids who can’t take an Algebra test to save their life, but they can create wonderful interactive documents that very nicely illustrate mathematical ideas. I keep thinking, THAT’S how their going to be using math in their life – they’re going to be MAKING things with it! And in making things, they’re not going to be doing the calculating. Yet, that’s what our curriculum still teaches and tries to measure – how well the kids can perform algebraic calculations on a random assortment of problems within a fixed amount of time.

I’d deeply appreciate any further thoughts you have along these lines. How do you approach assessment? How well has it worked to have ‘no tests, quizzes, exams, and no homework’? I think there’s something very important here.”

Michel’s email got me thinking about how important project work is to me in teaching mathematics, and how I might better articulate the principles I use in  assessing student project work.

If the biggest  handicap to using project work more often is that some people – mistakenly in my view – think this is somehow “lowering standards” then those of us who know how engaging project work can be for both students and teachers need to do a better job spelling out assessment procedures in greater detail.

That was my intention in writing this post, when I got another email this morning from Michel:

“… I had an interesting insight regarding how some kids come up with their projects. Monday was the beginning of our second semester, and I used the first class period to share the projects with the kids. When I displayed a project I would ask the author some questions about it. What I discovered is that You Tube has a whole bunch of GeoGebra tutorials! One kid had apparently done a really great interactive visual proof of the Pythagorean Theorem. I kind of suspected that perhaps he hadn’t actually come up with it on his own, because when I asked him some things about it, I saw that he didn’t understand that it was a proof! A few more similar situations, and I found out that You Tube was the source of many of the ideas. I found that amusing. I wasn’t terribly disappointed, after all, why not use resources that are available? But I realized that I did have to factor this fact into the mix. So, some of the amazing creativity I was seeing might not have been original. But some of it definitely is. For example, one kid last year created a set of sliders that would spell out his name ‘Jason’ when moved. Each letter was controlled by its own sliders. He created the ‘s’ using two sideways parabolas. He asked me for a little help on the ‘s’, so I could see that it was his own.“

So now I had a different issue: is copying a bad thing? Copying something directly and passing it off as your own is plain dishonesty. A student who does this learns little to nothing about the mathematics involved. But what about copying with modification? That seems to me to be an excellent way to begin to figure out how something works.

What are the YouTube videos Michel mentions in the email?

GeoGebra has a GeoGebra Channel on YouTube, with many examples of GeoGebra applications.

Here is one of many very cute examples of GeoGebra applications on YouTube:

I would encourage younger students to take these examples and pull them apart: modify them, play with them, change them to do something else. That way they learn a lot about how GeoGebra works, they learn some useful mathematics, and they have fun.

In a later post Michel and I, and one of Michel’s students, will discuss projects like these, their assessment, and how they help students develop mathematical proficiency while having fun.

Reuters new agency reported on January 20, here 2011 that a Japanese systems engineer, Shigeru Kondo, had used a home-built computer to calculate to  trillions of digits and thereby  broken the existing Guinness Book of Records entry for calculating .

A sentence toward the end of the report caught my eye:

“Calculating a more accurate pi, which is believed to go on forever, has been a challenge for scholars for thousands of years, ever since the parameter was used in ancient Egypt.” (My emphasis)

is not a rational number

Of course the decimal expansion of goes on forever, because is not a rational number. That is, there are no integers for which .

This was first proved by Johann Heinrich Lambert in 1761.

The irrationality of means not only does the decimal expansion of go on forever, but that it never repeats from some point on.

Of course we are familiar with rational numbers whose decimal expansions repeat, such as , and which repeats after the first two decimal places.

Equally, any decimal that eventually repeats is a rational number.

The reason for this is fairly straightforward.

First, if is a number that has a repeating decimal after, say, the decimal place then is a repeating decimal.

Second, repeating decimals are rational numbers – just multiply by where is the length of the repeating block, to get .

For example, if then and if then so . So is a rational number, and so therefore is .

All this means that not only does the decimal expansion of go on forever, it never eventually repeats.

is transcendental

The same is true, of course, for which is an irrational number.

What distinguishes from a number such as is that the latter is algebraic.

A number is “algebraic” if there is a polynomial

all of whose coefficients are integers, for which

Of course, in the case of such a polynomial is .

For there is no such polynomial. This is what is means for to be a transcendental number: is not algebraic.

This was first proved by Ferdinand von Lindemann in 1882.

Algebraic numbers are in many respects easier to understand than non-algebraic (= transcendental ) numbers. In fact, Leopold Kronecker denied the existence of transcendental numbers (including ).

Are the digits of random?

This is a very difficult question to answer unless we have a workable definition of “random”.

Gregory Chaitin and Andrey Nikolaevich Kolmogorov came up with a definition of “algorithmic randomness” which has to do with the non-compressibility of the description of a number. While this is an interesting and useful idea it does not seem to correspond exactly to what most people think of as randomness.

In practice, a sequence of numbers – such as the digits of – is NOT random if it fails one of a batch of tests for randomness.

What this means in practice is that we cannot prove that a sequence of numbers is random, only that it is not random.

If a sequence of numbers passes all known tests for randomness we tentatively conclude that the sequence is behaving like a random sequence.

A 2001 research report by Paul Preuss at Berkley, indicates that the sequence of digits of does behave as if it were random.

So not only, do the digits of go on forever, they do not eventually repeat, and appear to all intents and purposes, to be random.