tag:blogger.com,1999:blog-1907702537884089718.post1515431442871419200..comments2017-03-28T14:07:06.741-07:00Comments on Exit 10A: It's Never Too Late to LearnJoe Schwartznoreply@blogger.comBlogger6125tag:blogger.com,1999:blog-1907702537884089718.post-16503342696556921692016-09-17T14:58:49.382-07:002016-09-17T14:58:49.382-07:00Thanks Kent! I'm just a beginner here, but th...Thanks Kent! I'm just a beginner here, but the GGI book has influenced my practice in similar ways. I had many "a-ha" moments, and, like you, I love the way they introduce fractions with equal sharing problems beginning with mixed numbers! Looking back in my book I wrote "Wow!" and "Duh!" in the margin right there on pg. 6. It makes so much sense. In terms of formal fraction notation, perhaps that's why the CC holds off on that until grade 3. I also agree that backloading the vocabulary is the way to go. Interesting: a common thread is that these practices are counter to the way things have traditionally been done.Joe Schwartzhttp://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-23420821846529800642016-09-15T11:02:36.401-07:002016-09-15T11:02:36.401-07:00For one, the book (and CGI in general) taught me t...For one, the book (and CGI in general) taught me that kids do have ideas about topics that they haven't been formally instructed in, and it's better to figure out those ideas and try to build off of them instead of ignoring all the thoughts that kids bring with them into the classroom. <br /><br />Another thing I love from this book is the way they show how formal fraction notation, like 1/3, can hamper long-term understanding if it's introduced too early. I LOVE the way they introduce fractions using equal-sharing problems and then start talking about halves, thirds, and fourths as units, the way we would talk about inches and feet and miles. Obviously you can't just add a third and a fourth, in the same way you can't add an inch and a foot, without converting them in some way. <br /><br />I would say that I now backload my units with vocab and notation in a way that I hadn't in the past. For example, in my first unit of Algebra, I ask kids to describe the domain and range of functions, but I don't require interval notation. I just want them to say "everything less than -3" or "from 4 through 11" in words. Then once they have that big idea down, I can introduce interval notation (probably in unit 2, when we solve and interpret inequalities explicitly.)<br /><br />Lastly, the book very delicately tells teachers that they may be completely screwing up the way that we introduce fractions to students. It has made me wonder what other topics in math are just sequenced completely incorrectly.<br /><br />For example, I don't think that we need to ever teach one-step equations in middle school. To me, the idea of reversing the order of operations makes MUCH more sense to kids if we start with multiple operations. So I don't even touch one-step equations as a topic anymore. I start with two- and three-step equations, and I think kids get it a lot better. One-step just fails the reasonableness test for most kids. "I know that x is 4 when 3x=12. Why do I have to divide both sides by 3 to prove it? This is stupid." But if I say "Joe picked a number, multiplied it by 3, added 5, and then subtracted 7. He ended with 28. What did he start with?" The whole problem just makes much more sense to kids, and working backwards seems natural.Kenthttp://www.blogger.com/profile/17251777011355228189noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-76822493336899583982016-09-15T02:31:11.308-07:002016-09-15T02:31:11.308-07:00Thanks Kent. I'd be curious to know how you f...Thanks Kent. I'd be curious to know how you feel it's informed your practice.Joe Schwartzhttp://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-47120993434304604972016-09-15T02:30:28.327-07:002016-09-15T02:30:28.327-07:00It seems that CGI is prescribing a pedagogical app...It seems that CGI is prescribing a pedagogical approach based on their theory of how children think. I do have the new edition of Children's Mathematics, but haven't read it yet. Do you mean it's less cautious in offering pedagogical prescriptions than the first edition? Is the something lost the teacher's autonomy to deviate from the CGI prescription? Joe Schwartzhttp://www.blogger.com/profile/02304083254248927187noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-11060828142833146012016-09-14T18:48:01.219-07:002016-09-14T18:48:01.219-07:00Still my favorite math book I've ever read.Still my favorite math book I've ever read. Kenthttp://www.blogger.com/profile/17251777011355228189noreply@blogger.comtag:blogger.com,1999:blog-1907702537884089718.post-78125402751427316282016-09-14T17:16:52.808-07:002016-09-14T17:16:52.808-07:00Is CGI a pedagogy, or a theory of how children thi...Is CGI a pedagogy, or a theory of how children think?<br /><br />From your post, and the fractions book:<br /><br /><i>"The new role centers on helping students communicate strategies to other students, directing questions to specific students to help them draw connections between these strategies and more basic strategies, introducing equations to represent students' strategies, and highlighting the fundamental properties of operations and equality that that underpin these strategies."</i><br /><br />From the 1st Edition of Children's Arithmetic:<br /><i>"During this conversation, I realized how serious [the researchers] were about respecting teachers' judgments on particular issues. Since they had little evidence about representing these situations, they would see how teachers and children handled it. As they worked with teachers, sharing their research knowledge about students' learning of addition and subtraction, they would continue to learn from teachers and children."</i> <br /><br />So there's something cool here, a shift in how CGI has presented itself over time. It's gone from resisting pedagogical prescriptions to offering them freely.<br /><br />(The <a href="http://rationalexpressions.blogspot.com/2014/09/cgi-is-for-high-school-teachers.html" rel="nofollow">new edition</a> of Children's Mathematics is far less cautious. To me, that's something lost.)<br />Michael Pershanhttp://www.blogger.com/profile/17046644130957574890noreply@blogger.com