tag:blogger.com,1999:blog-8850230383845297552.post75162112555573381..comments2017-09-19T11:53:05.925-07:00Comments on Fail Early and Often: AutomathographyAndy Pethanhttp://www.blogger.com/profile/05159258049094512496noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-8850230383845297552.post-18477493493688536302014-08-20T09:53:25.310-07:002014-08-20T09:53:25.310-07:00Wow -- thanks for all of the thoughts and question...Wow -- thanks for all of the thoughts and questions! Here are some of my responses:<br /><br />The good and bad of early-age acceleration: I actually didn't see any major downsides to accelerating. If I was instead allowed to go at the pace I wanted to instead of being constantly slowed down, I think I would have hit the "challenge" stage much sooner than Calc and developed a healthier attitude of math as a place to explore, not a place to "win". I wish more schools were setup in a way that would encourage more students to work ahead and explore at a younger age.<br /><br />How does my peer/self learning affect my teaching?: In stats, where most of my curriculum development efforts have been, I try to design experiences that make it easier to discover big ideas and make sense of them with explicit instruction from the teacher. That said, I've created hundreds of videos that are almost all direct instruction in the past 3 years. With these, students choose the pace and whether or not they need them, which I also value, but they don't require an inquiry approach to learning. Though I'm too far on the pendulum towards direct instruction, a balance is needed.<br /><br />Calc -- the top of the ladder?: Perhaps I'm too into stats these days, but I see high school math diverging down two paths -- algebra/calculus and statistics/modeling/simulation. In college, as an engineering student, I didn't use almost any analytical techniques to solve calculus-type problems, instead using computer simulation. Most careers, including many techy careers, use tons of stats and little algebra/calculus. Thus, I wish we would stop treating it like it is so special and push more kids towards stats. So that's how I personally treat it.<br /><br />Do I still compete in math? What is an example of a nuanced understanding I now have?: I do not thankfully feel like I'm competing in math anymore. Part of that is that I am not chasing new frontiers / topics, but instead working within to re-think and build new connections between topics. In a competition, you want to check off boxes fast. In an exploration, you want to see the details and look closely at things in new ways. One example of this is my realization that addition is repeated counting, multiplication is repeated addition, and powers are repeated multiplication, and that forms the basis of most of our operations. I think I will try to writeup a blog post that expands more on this soon.Andy Pethanhttps://www.blogger.com/profile/05159258049094512496noreply@blogger.comtag:blogger.com,1999:blog-8850230383845297552.post-4152360323001006272014-08-16T19:10:55.657-07:002014-08-16T19:10:55.657-07:00I have definitely had students who were like you a...I have definitely had students who were like you and tried to write down the minimum necessary to answer the problem in teeny tiny handwriting. I think it's interesting that you ended up preferring doing math on the whiteboard. I'll have to keep that in mind with for my students. I like your comment at the end about using our imaginations. I'll remember that this year as I'm teaching.Mary Cumminshttp://marylikesmath.wordpress.com/noreply@blogger.comtag:blogger.com,1999:blog-8850230383845297552.post-73352102988476650262014-08-15T19:08:30.092-07:002014-08-15T19:08:30.092-07:00Thanks for sharing your story, Andy.
I had a simil...Thanks for sharing your story, Andy.<br />I had a similar experience as you—math was something that helped me to forge social connections with my peers that was difficult for me otherwise. A big example of this was as a high school freshman, when I did a lot of peer tutoring after school in my teacher's classroom.<br />Looking back on your acceleration in math at an early age, what was good about it and were there any downsides? How do you feel about acceleration for your own students?<br />Something that comes up several times in your story is you figuring things out yourself or with a group of peers—rather than being taught or guided through math by a teacher. That's really interesting. Does that ring true to you? Is it the same or different from others' experiences? How does your past in this respect affect your own classroom and your attitude towards learning math?<br />I’m glad you had a patient teacher when you had all those thoughts about multiplying fractions to resolve!<br />“Logic didn't feel like math to me, but was instead a challenge in organization and argument.” I’ve encountered this thought pretty often from kids, and I don’t fully understand it. I’d be interested in teasing apart the difference in feeling of “math” and “logic”.<br />Graph sketching is the worst. :)<br />“Calculus. I hardly knew what it was until I was in the class, but I knew I wanted to get to it.” On the one hand, I suppose this is perfectly natural, but I also think this pretty standard feeling is also a little bonkers. Do you remember Calculus living up to your expectations? How would you talk with a kid about Calculus (as a subject, as the “top of the ladder”) to a younger high school student who was in a similar position as you were?<br />Your experience with whiteboards is really cool. I appreciate your perspective on collaborative work, both here and in your elementary years and in what you say later about the year you took off from college.<br />“I still learn best by seeing lots of examples and drawing my own generalizations and conclusions.” What are some other ways that people learn math best?<br />Your DiffEq experience is a bummer to read about. It sounds like the class was poorly set up. I’m interested, though, in the arc you’ve traced. It seems like math was easy for a long time for you. And then Calculus was a challenge, and then DiffEq was a wall that shut you down. I feel like the basic shape of this experience is a pretty common one, though for some people the wall is fractions and for others it’s Algebra 1 or Linear Algebra. Which is a bummer and, on some level, so tragic—because where’s the support? It’s almost as though this is the way math education is *supposed* to work.<br />Do you still feel competitive about math? How does this—or the change in this—shape your experience of the subject as an adult?<br />I had a similar experience in figuring out ways to get into the classroom as a teacher as quickly as possible after undergrad. It’s what led me to independent schools, where I didn’t need a credential.<br />“I'm finding that many of the limitations are not from "the system", but from our lack of imagination on how things could be.” Love it. Who we are as teachers is the biggest ceiling on what can happen in our classrooms. It’s a huge challenge and undertaking, but we’ve got to find great resources and colleagues and collectively and individually pull ourselves up by our own bootstraps.<br />What’s an example of a “deeper, richer, more nuanced understanding of math” that you’ve gained so far?<br />I’m glad for the happy accident of you becoming a math teacher! Thanks again for sharing, Andy.Justin Lanierhttps://www.blogger.com/profile/11386367931599418555noreply@blogger.com