Math Relearning/Progressions/K-5 Measurement
These are my comments on the Common Core Math Progressions document Measurement & Data (measurement part): Grades K-5 (PDF).
Ah, they agree with me on the centrality of measurement.
These math educators always contrast measurement and counting, but I want to highlight their similarities. They both use units, for example.
The Standards don't distinguish between mass and weight. Shame, shame!
"direct comparison" - My earlier sources annoyingly didn't explain that direct comparison was meant to happen without measurement as a step on the way to measurement, so I was having trouble fitting it into my logical map of measurement concepts.
The list of measurable attributes in this section reminds me of Thad Roberts' TEDx talk where he reduced all the units to five basic ones: length, mass, time, charge, and temperature (http://einsteinsintuition.com/what-is-qst/constants-of-nature/).
One of the recurring questions of my life: Why are there multiple measurable attributes that an object can have? Why isn't everything just length or something?
"comparing it to the amount" - This corresponds to my idea that measurement is a ratio. It also hints at the idea that at a certain point, the magnitude of a unit has to be established by direct experience if we're going to make any sense of it. A numeric measurement doesn't mean much on its own.
"object is subdivided" - It's interesting to contrast this with division.
"parallels the number concepts" - Good point comparing the metric system to place value.
"Scientists measure" - Fair enough.
They don't mind using length as a synonym for distance, and they explicitly define it so it can be. They even describe volume as entailing three lengths simultaneously.
They agree with me that length is a core concept and for the same kinds of reasons I was giving.
Area - This is where we get into things I hadn't thought about yet.
How do we know conservation is true? In any case, it has nuances. And I think the concept is somewhat different between math and science.
Seriation - A good opportunity to teach sorting algorithms. :)
"no gaps or overlaps" - I have some thoughts on this, which I call coverage, in my measurement notes.
inequalities - Impressive. Must've been a gifted student. :P
paper-folding - Interesting about congruent parts. It can teach other math concepts too! Though maybe not in grade 1.
"one-dimensional unit structure" - Yes, many measurements are conversions to length.
"begin counting" - Off by 1! But yes, even rulers have semiotics. I vaguely recall being confused about the start-or-covered question in some counting situations with some tool or other. Actually redstone power in Minecraft is one.
"units of different sizes" - This reminds me of the scene in Spirited Away where the girl had to hold her breath as she was crossing the bridge. The commentary said this was to illustrate the childhood experience of having to follow the seemingly arbitrary rules adults establish. Hence, I note that when kids try to guess the rules, they get it wrong.
accumulation - Interesting, and that wasn't in my earlier sources or thinking.
zero-point - Measurement is relative to a reference point.
"length-unit size" - Yes, in counting, the unit size is one object. This is true for ordinal numbers too.
"inverse relationship" - This would add a level of unpredictability that would make me anxious to find out if there was a way to predict the measurement given the unit size. What button is it pushing for me?
regular vs jumbo paperclips - Another example of how math concepts can be mixed and matched to reveal more relationships and avenues to knowledge. So what is it about units that's being reapplied, and how? I feel like notating the concepts with some kind of symbolic shorthand would help with the analysis. So "the larger the unit, the fewer number of units in a given measurement" means something like, "if unit[a] > unit[b], num[a] < num[b]." Then "if object A is 10 regular paperclips long and object B is 10 jumbo paperclips long, the number of units is the same, but the units have different sizes, so the lengths of A and B are different." This means "If num[a] = num[b] and unit[a] < unit[b], then len[a] < len[b]." So we're using transitivity. Hmm, sort of. Anyway, it's good to keep in mind the phrase, "So with that, we can ..." and look to see where we can apply pieces of the concepts we've learned.
"benchmark lengths" - Find lists of these. Maybe EngageNY will have enough.
multiplying side lengths = counting tiles - Yes, and why they're the same as scaling.
"areas are preserved under rotation" - This is true physically, at least with solids, but why is it always true mathematically? Maybe it helps to think of numbers as rigid like solids. They always take up the same (quantitative) space.
independence of area and perimeter - How does that work? With rectangles it's because the sides of the square units can be part of the edge or interior, depending on their arrangement.
perimeter formula verbal summaries - It would be interesting to spell out the steps in translating the verbal summaries into the formulas and vice versa.
"emphasizes the step" - I hadn't thought of using particular formulas as memory aids. Formulas can have other kinds of advantages, such as illustrating the distributive property or reducing the number of calculations (see earlier).
"How long is the garden?" - Sometimes I read word problems and wonder, "Why would that be unknown?" One set of cases would be if you were gathering the data from different sources or occasions. One piece could've been collected for one purpose and another for another, and now you have enough information to learn more by calculating instead of measuring.
Coming up with situation equations involves knowing which operations are appropriate for the numeric relationships within the situation. What tells us that, for example, perimeter is additive?
"concepts of angle" - This is similar to what I'd been thinking about angle but hadn't written yet. I like the "change in direction" language. I think I was tying angle to the difference between circles and shapes with vertices, which I thought of in terms of the discrete-continuous distinction. Circles (or ellipses) are made of a line that changes direction continuously, whereas vertex shapes (is there a name for this category?) are made of a line that changes direction discretely. I'm sure there's a more mathematically correct way to say that. I was contrasting angle with distance. Angle is a type of distance but a rotational one rather than linear. Explore the nature of this distinction.
The different ways of describing angles reminds me that I've seen different ways to define a circle. Explore the relationships among these definitions.
Why do we use 360 degrees?
Why are 90 degree angles special, and why are they called right? They might be special because they're half a straight angle, which is just a line. I was thinking it was because they're symmetrical when you rotate them or some such thing.
I'm looking forward to geometry, by the way. There's something about space that's so interesting and inviting. Plus I like visual things.
What's a relational way to define angle? Rotation sounds operational. Maybe the difference between two directions. That still seems to have rotation in the background, since you could try to measure the difference linearly, but maybe there's a relational way to define rotation.
"angles in a variety of situations" - Ah, yes, angle is an abstraction of these kinds of situations.
How would a blind person learn geometry?
angle of turtle rotation vs angle formed - What? Guess I'd have to see it.
Why a turtle? Maybe because its legs form crosshairs and its head tells you which direction it'll be moving.
What makes me want to skip over the examples on p. 25? I'm going to try not to skip exercises and to ask that question a lot when I'm going through the actual curriculum. In this case, and I suspect in many, it's the mental load of remembering numbers and relationships. Being able to write things down would make it easier. I can with the pencil tool in my PDF software.
Another thing that bothers me about math problems is sorting out the path to the solution when you have a lot of pieces to coordinate, like in this example. Again, writing might help.
"Convert like measurement units" - How do fractions and decimals relate? I know the procedures. I just want to trace what happens, because now that I think about it, it seems strange to me. How does division turn a fraction into a decimal?
Kumi problem - The diagram reveals something I wouldn't have thought of, that half of what remains after subtracting 1/5 is 2/5, so to find the value of 1/5, just divide by 2. Now that I say that, it's interesting to me that one number can represent another number. What's happening there?
decompose right rectangular prism - For the three axes I was just imagining someone bending over at a right angle with their arms outstretched, or maybe held to the sides and bent outward at the elbows.
All that partitioning is awfully organized. What would measurement be like if we were partitioning into irregular, organic shapes? Why do we need regular intervals for measurement? It's obvious that we do, but I want to see it expressed in case I learn anything. I just want to know why we privilege square-related shapes in geometric measurement. But maybe there are other types of measurement that use other shapes, such as angles, which use circles. See? I noticed a relationship.
packing and filling - Packing depends on the fact that volume is additive.
I have http://www.twitch.tv/bobross playing in the background, sometimes looking and laughing at the chat. Then I return to this somber math reading. I wonder what makes a math joke. There are puns, but I wonder what other kind of math humor is possible. This could be important for education.
Where the Geometric Measurement Progression is heading
The geometry progression does seem kind of slow.