A very quick primer, to help you locate me within the structure of the Elementary Ed program: The full year is divided into several terms. The fall semester consists of Term II and Term III. Right now, I’m approaching the end of Term II, which focuses on learning from an individual child within our placement classrooms. (Term I focused on the school in its neighborhood; the next term, Term III, focuses on working with small groups; in the spring, we focus on whole-class instruction.)
For the end of Term II, we have an “integrated assignment” focused on that child, which pulls components from most of the classes we’re currently taking:
I did, however, do the math task with my student, and man oh man was it interesting. In fact, let me use that as a springboard to tell you a bit about my Math Methods class, which I find fascinating mostly because of how different it is from my experiences with math instruction growing up.
--Warning: digression about Math Methods to follow. Probably most interesting for people who want to learn about the class, who love math, or who hated math in school and want to know why--
Suppose I asked you two word problems. In the first, Lucy has 21 cookies and wants to put them in bags of 3 cookies each; how many bags can she fill? In the second, Luke has 21 chocolate chips, and wants to divide them evenly among his 3 cookies. As an adult, those problems are both easy, and they’re both identical: 21/3=7. And, in my experience, my teachers agreed with that, and would simply correct me by reminding me that they are both division problems.
For someone who is still learning about division, however, those problems could look totally different. To solve the first one? Maybe it’s easy, you count up by threes until you get to 21, then notice you’ve counted 7 threes. But the second one? The process might be a lot less instinctive – do you count down from 21, mentally putting one chip at a time in each cookie? Or do you put some in each cookie, then see how many remain, then divvy those up?
Another example: do 29 + 86 in your head now!
How did you do it? Did you actually line up the problems and carry the one like you would using a “standard algorithm?” Or did you know immediately that 20 and 80 are 100, and then tack on the rest? Or did you bring over the 6 to get 35 and then add on the 80?
The key takeaways from these two cases are that context matters a lot, and that students relate to math problems in complex and individual ways. Rather than teaching math as a series of rote facts to memorize (which is not how most kids’ brains work), math should be taught in ways that value deeper understanding of the relations at work within math problems, and that helps students develop their own strategies and shortcuts for problem solving.
Which is fascinating. Right? Right?
On the one hand, this seems obvious; on the other hand, it’s very different from my memories of learning math. Yeah, I remember playing with unit blocks and 24 cards, but in this understanding of math teaching, one of the most valuable tools is having Math Talks, where students share their diverse solutions to simple problems to help expose students to a variety of ways of thinking about mathematical concepts (not to mention build a tool-kit for problem-solving).
So…back to me.
Like I said, I did the math portion of my integrated assignment, and it was totally interesting! For all those reasons I describe above. My student got every problem right – but his processes and explanations were often completely different from what I would have expected. In some cases, he nailed the solution but showed no deeper understanding of the math principles involved; in other cases, he showed off some extremely complex understandings of numerical relationships.
If you can’t tell: this kind of thing makes me a bit giddy. Which helps me feel pretty confident in my choice to become a teacher. A feeling that’s always welcome, to be sure.
For the end of Term II, we have an “integrated assignment” focused on that child, which pulls components from most of the classes we’re currently taking:
- For our Math Methods class, we observe the child working through a series of problems, and try to interpret and assess their modes of understanding
- For Science, we work with the child to carry out an experiment on sinking and floating
- For English, we practice assessing literacy while also interviewing the child about their personal relationships to reading and writing
- For Social Studies, we record and “map” the student’s physical use of the classroom space (an assignment that seems tied to this course more for convenience than connection)
- And for our Field Seminar, we tie it all together with a preceding descriptive review and a concluding reflection.
I did, however, do the math task with my student, and man oh man was it interesting. In fact, let me use that as a springboard to tell you a bit about my Math Methods class, which I find fascinating mostly because of how different it is from my experiences with math instruction growing up.
--Warning: digression about Math Methods to follow. Probably most interesting for people who want to learn about the class, who love math, or who hated math in school and want to know why--
Suppose I asked you two word problems. In the first, Lucy has 21 cookies and wants to put them in bags of 3 cookies each; how many bags can she fill? In the second, Luke has 21 chocolate chips, and wants to divide them evenly among his 3 cookies. As an adult, those problems are both easy, and they’re both identical: 21/3=7. And, in my experience, my teachers agreed with that, and would simply correct me by reminding me that they are both division problems.
For someone who is still learning about division, however, those problems could look totally different. To solve the first one? Maybe it’s easy, you count up by threes until you get to 21, then notice you’ve counted 7 threes. But the second one? The process might be a lot less instinctive – do you count down from 21, mentally putting one chip at a time in each cookie? Or do you put some in each cookie, then see how many remain, then divvy those up?
Another example: do 29 + 86 in your head now!
How did you do it? Did you actually line up the problems and carry the one like you would using a “standard algorithm?” Or did you know immediately that 20 and 80 are 100, and then tack on the rest? Or did you bring over the 6 to get 35 and then add on the 80?
The key takeaways from these two cases are that context matters a lot, and that students relate to math problems in complex and individual ways. Rather than teaching math as a series of rote facts to memorize (which is not how most kids’ brains work), math should be taught in ways that value deeper understanding of the relations at work within math problems, and that helps students develop their own strategies and shortcuts for problem solving.
Which is fascinating. Right? Right?
On the one hand, this seems obvious; on the other hand, it’s very different from my memories of learning math. Yeah, I remember playing with unit blocks and 24 cards, but in this understanding of math teaching, one of the most valuable tools is having Math Talks, where students share their diverse solutions to simple problems to help expose students to a variety of ways of thinking about mathematical concepts (not to mention build a tool-kit for problem-solving).
So…back to me.
Like I said, I did the math portion of my integrated assignment, and it was totally interesting! For all those reasons I describe above. My student got every problem right – but his processes and explanations were often completely different from what I would have expected. In some cases, he nailed the solution but showed no deeper understanding of the math principles involved; in other cases, he showed off some extremely complex understandings of numerical relationships.
If you can’t tell: this kind of thing makes me a bit giddy. Which helps me feel pretty confident in my choice to become a teacher. A feeling that’s always welcome, to be sure.
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