MAT: Unreal Decimal Points

Back in late September, Christopher Danielson, aka Triangleman, created the "Triangleman Decimal Institute", a six week course looking into the whys and whens of decimal instruction. In this seventh week, he's asked a single question: How can you show the world what you have learned these last several weeks?

Stick around through my mini-summaries, at the bottom I talk about calculators.

WEEKLY ROUNDUP

1: Decimals before fractions? Often the decimal (or at least notation) is taught before fractions. Perhaps with the idea that they are more like whole numbers. I think in the end, I would agree that they are more like whole numbers, particularly in terms of place value, but are different enough to be a problem.

2: Money and decimals. Money is often used to teach decimals. Is this valid? I would say no - dollars and cents are seen by people as two separate units, not an extension down into "parts of dollars". I think something that exacerbates this is that the groupings are different... you have 25 cents, but 20 dollars (unless you're talking Euro). I wonder about the origin of the quarter.

3: Children's experiences with partitioning. Real world knowledge that children bring into classrooms - it tends to start with cutting in half, then half again. Possibly even throwing out a quarter to make things fair between three people. I wonder whether thirds are seen as inherently unfair, because they don't have a "nice" decimal representation (in money or otherwise)? Though I've no means to test this.

Just found this similar 12-slicer on the internet.
Don't know if that's good enough.

4: Interlude on the slicing of pizzas. Direct documentation of pizzas. I was busy that week, though I learned that I couldn't document a 12 slice pizza, because I had no technology that allowed for instantaneous photographs during my Saturday game night. Also, places in Europe don't serve pre-cut:  you choose your own slicing.

5: Grouping is different from partitioning. The idea of starting with parts and combining them, versus starting with a whole and breaking it down, aka moving the decimal to the left or right, aka what makes a "1" (unit). I agree there's a difference, and I don't think decimals come naturally from partitioning.

6: Decimals and curriculum. Common Core State Standards were mentioned in the United States. Similar to Ontario, decimals are introduced in Grade 4, but there they immediately use up to two decimal places. In Ontario, we have one decimal place only, and gradually add another as they move through the subsequent few grades.

FINAL THOUGHTS

I started by thinking decimals were subsets of fractions. I'd now say decimals are a SCIENTIFIC measure... which is corrupting the study of mathematics.

I'm overdramatizing there, but bear with me. A few people (in Week 5 in particular) remarked that students preferred fractions over decimals. Not necessarily that they preferred fraction OPERATIONS, merely use of fractions. And I think that's what children do when they partition, they break things down in a fractional, more "fair" way. Fractions (pieces) are, in a sense, more natural than decimals.

Units! We... wait, what?

If you're starting with "1" unit, you have "one half" or "one third", which is SEEN right in the notation! (1/2) You don't have ".5". You might have "0.5", which finally mentions the whole, but that puts us in the realm of significant digits and scientific notation. Which is more like degrees of magnitude, not parts of a whole. (Magnitude being something we're not good at - consider this ViHart/Sal Khan video.) Same problem with money, we have two places by definition - unless we're talking about gas prices, which can lead to that crazy picture (in New Jersey) offered up by David Wees.

Granted, I bring some of my own bias into these thoughts. I'm reminded of a conversation I had with a student a couple weeks ago, about an exponential model with an asymptote at 21 degrees. They said that the item WOULD eventually be 21 degrees, so how could this be an asymptote? My counterpoint was that it was technically 21.0000000000001 degrees, but our tools cannot be that precise. (Nevermind that a uniform 21 degrees in a room is likely impossible.)

The thing is, while fractions are more natural, decimals are our reality. Why? Calculators. More specifically, SCIENTIFIC calculators.

These tools work with place value, as they were designed to do. Something that works the same either side of the decimal point, and allows for expressing an answer in scientific notation. (Aside: Which few students understand... what's with 'E-7'?.) Now, can you imagine the difficulty of programming fractions into early calculators?? The idea of incorporating fractions into a calculator came later. Most computer calculators STILL don't have a fraction key!!!

My calculator's in 'P' mode.
How do I fix it?

But that's fine. Calculators do what they were designed to do. The trouble is, what they were designed to do was reduce everything down to decimals. In using them, decimals have become our reality. The fact that we seek out "real world" applications reinforces that. So am I saying DON'T use calculators? Or for that matter, don't use the metric system? Heck no. I'm saying be aware of the following: More numbers exist that CANNOT be expressed using decimals/fractions than numbers that do.

We don't generally care about those numbers until Grade 11. By that point, decimals might be the default. So what's the notation for half of root 2? Is 0.5root(2) sufficient? Is that sort of division even "fair"? How easy is that number to estimate? Do you even care?

If you don't care, I ask: Is mathematics the study of numbers, or is it the study of "real world applications"?

I don't have answers to those questions. Of course, as any scientist would say, I don't think all the evidence is in yet. So that's what I've learned.