tag:blogger.com,1999:blog-1907702537884089718.post8349707048029406578..comments2018-07-18T10:38:01.157-07:00Comments on Exit 10A: In OrbitJoe Schwartznoreply@blogger.comBlogger14125tag:blogger.com,1999:blog-1907702537884089718.post-50692779935508317282018-01-09T01:16:34.373-08:002018-01-09T01:16:34.373-08:00Excellent post.Thanks for sharing this message.Rea...Excellent post.Thanks for sharing this message.Really,Your blog is not only instructive but useful too.<br /><a href="https://www.learninghouse.ca/category/9/TLH/math.html" rel="nofollow">Math Curriculum</a><br />John Peterhttps://www.blogger.com/profile/07699146535492722078noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-66059046026939853592017-05-28T10:04:45.070-07:002017-05-28T10:04:45.070-07:00I didn't think of it that way but yes, that...I didn't think of it that way but yes, that's a useful way for me to think about it!Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-55212045599137704542017-05-28T06:14:36.674-07:002017-05-28T06:14:36.674-07:00"What does the student who labeled the pieces..."What does the student who labeled the pieces as a count know about fractions?" <br />As I was writing the post I had a conversation with Graham Fletcher about this, and that was the exact same question he asked. He argued that you could label the pieces as a count and still understand that each piece was still only 1/16 of the whole. But I'm not even sure that the categories apply to fractions, although it seems they they must.Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-58649213234625379172017-05-28T06:08:13.158-07:002017-05-28T06:08:13.158-07:00So it's not numbers but uses of numbers that c...So it's not numbers but uses of numbers that can be classified?Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-75258382077135697112017-05-28T06:04:35.619-07:002017-05-28T06:04:35.619-07:00I agree. I think that context has to be taken int...I agree. I think that context has to be taken into consideration. Just a naked number can't be classified without context, though I think most times a naked number is taken to be cardinal.Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-40832659203596916552017-05-27T09:46:04.334-07:002017-05-27T09:46:04.334-07:00The clock example had my wheels spinning too! As ...The clock example had my wheels spinning too! As I think about it's classification, it feels like it could be cardinal or ordinal depending on the context. If I were considering "how many hours it has been since _____" or "in how many hours it will be _____", that feels cardinal. But when I think about the 11th hour, the context might lean more toward ordinality. I am now thinking that all the numbers we see must be considered within context to classify them. Thoughts?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-87701196360013644222017-05-26T06:33:26.849-07:002017-05-26T06:33:26.849-07:00This was a great read and my thoughts are spinning...This was a great read and my thoughts are spinning. I will likely be trying to classify numbers all day. <br /><br />I remember working with number line vs number ladder this year and trying to classify which one would represent ordinality and which one cardinality. A ladder consists of spaces, no zero, numbers are like counting blocks occupying space. A number line can be infinitely broken into numbers and makes me think of more of a number river than number path. It also has a zero. I thought of a number line as representation of cardinality, where numbers do contain within them the previous numbers. And ladder seems to represent number blocks, the counting step,ordinality. But then counting objects is actually ordinal relationship, not cardinal.<br />We were thinking about the time with my students, is time and calendar more like spaces or lines? We thought about ordinality of our calendar without zero, with days and years like blocks. But now after reading this post and the comments I realize that the units for our time measurement are nested within each other. There is a modular feel to them, but the minutes don't just reset every hour; they switch gears for hours, hours push days to move, days reset to change months and 12 months start over when we add one to the year count. Like Sarah mentioned, it actually seems like a place value system. I wonder if the time itself has the features of cardinality but the way we capture it is an ordinal system?<br /><br />Time to go to work and to sort numbers. This conversation is fascinating, I'd love to continue it.<br />TAnnalethttps://www.blogger.com/profile/02611816015421905358noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-46720416259550751512017-05-26T04:39:44.346-07:002017-05-26T04:39:44.346-07:00I love this post! I wrote a post in the beginning ...I love this post! I wrote a post in the beginning of the year about how I wonder if third graders re-visit the counting and cardinality standards when they begin to study multiplication (https://mathontheedge.com/2016/11/18/one-what/) Now, you've got me thinking about fractions. It is like every time the unit changes, we have to revisit what we know about how the units fit together and what the last unit tells us about the rest of the unit. I am still wondering about the sixteenths. What does the student who labeled the pieces as a count know about fractions? And the modular conversation is fascinating. I have to revisit modular numbers, but I am wondering about the "next unit". So after 12 hours, we go back to the first hour, but those 12 hours now live inside a different unit called a "day". I absolutely love this conversation. I am going to go on a number hunt all day today. Sarah Cabanhttps://www.blogger.com/profile/04396651374828741208noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-33076070229796540412017-05-25T03:41:10.051-07:002017-05-25T03:41:10.051-07:00Thanks Simon! Yes, what I thought would be a pret...Thanks Simon! Yes, what I thought would be a pretty simple exercise turned out to be anything but. One thing I thought about but didn't explore is whether or not we could make any generalizations about the kinds of numbers used to measure things. For example, if you draw a line 7 inches long, you need all preceding inches, and it certainly includes 1, 2, 3, 4, 5, and 6 inches too. A box that holds 10 cubic inches holds all the preceding cubic inches as well. So those would be cardinal. Temperature is measuring, and so is time, so maybe they're cardinal too? <br />Agree we need an expert, which is why I'm hoping that Douglas Clements will chime in!Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-75034373622441733182017-05-23T13:07:36.370-07:002017-05-23T13:07:36.370-07:00I love this post Joe. I feel like you're diggi...I love this post Joe. I feel like you're digging into the meaning of these kinds of numbering - on the sort of level I, and, I guess, all of us need. Keeping it simple. What's an example of...<br /><br />So ordinal is like things in a line. Cardinal is counting things and saying the number is the last number you count, so that all the other things are inside like Russian dolls...<br /><br />Then you turn to the thermometer. The numbers are on a line, but do the temperatures include the lower temperatures? Does a comfortable 26°C day somehow *include* a chilly 3°C day? I guess if we have a physicist's view of temperature, we could say that the temperature represents movement; there's already some movement, then we add more movement on...<br /><br />I'd like to get really clear about this. I hope someone really expert comes along and explains it all in really simple terms for us!Simon Gregghttps://www.blogger.com/profile/07751362728185120933noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-91981103153857009272017-05-22T18:01:10.787-07:002017-05-22T18:01:10.787-07:00Sorry to bring back this painful memory, Dennis, b...Sorry to bring back this painful memory, Dennis, but thanks for sharing the experience. I like your description, "nominal with ordinal aspects." That's how I thought of the coat hook numbers. Maybe the Stanford folks could have used a little course on Gematria.Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-57639968636011718302017-05-22T17:56:06.105-07:002017-05-22T17:56:06.105-07:00Thanks Michael! I'm not even sure what modula...Thanks Michael! I'm not even sure what modular numbers are! But if they reset every time, then no way would 0 include 11. Time might function differently? Would counting the days in school be like that, too? 0 to 180 and then back to 0 again?Joe Schwartzhttps://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-36204512621682000552017-05-22T17:35:04.228-07:002017-05-22T17:35:04.228-07:00does each individual day include within it all pre...<i>does each individual day include within it all preceding days?</i><br /><br />Like the rest of this post, that line is pure poetry. I love this, Joe.<br /><br />The clock question is the one I keep thinking about. The thing that's bugging me is that clock numbers are like modular numbers. Can a modular number ever contain everything that comes before it? What <i>does</i> come before a modular number? <br /><br />If 11 + 1 = 0 in modular numbers, then does 0 include 11? No way, right?Michael Pershanhttps://www.blogger.com/profile/17046644130957574890noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-91050995147737907202017-05-22T14:44:57.896-07:002017-05-22T14:44:57.896-07:00You brought back one of my worst memories. In atom...You brought back one of my worst memories. In atomic orbitals, electrons filled them in an order of "s, p, d, ..." To me those letters were numbers. (Think Hebrew.) They were nominal with ordinal aspects. When I mentioned this to the professor and several grad assistants, they snickered at me and said I was misunderstanding: "Letters aren't numbers. Are you nuts?" I persisted because I was stunned people not only didn't know cardinal vs ordinal vs nominal, but directed hatred to the very "waste of time" in mentioning such a worthless concern. I was crushed because I thought I find at least one "smart" person who would agree, even if bored. Not one, and this was Stanford. A low point for me.Dennis Ashendorfhttps://www.blogger.com/profile/05099727076265177042noreply@blogger.com