We're now at the end of the fourth week of the winter quarter at the university and have settled into the new routine, though I've now added a new job to my list: I'm student teaching math to fifth-graders from extremely low-income and otherwise unfortunate/tragic family backgrounds. It's a very different experience from homeschooling, obviously, but the pedagogical tools and the variety of mathematical approaches I've learned from homeschooling have been absolutely invaluable. I've worked with two small groups of students, one at the just-below-average skill point and one well-below-average, and covered different topics with each. On Monday, the first group worked on simplification of fractions. Most of the students were fairly strong on their lower multiplication tables, but one didn't actually know that multiplication is just serial addition. Once he knew that, he could (slowly) solve anything I threw at him. I'm letting him play with that for a while (he's sorted out that counting by twos and threes is equivalent to and faster than adding them over and over--well done him!), and I plan to start working on the multiplication tables with him this week. It's well-timed, actually, because Oscar's going to begin memorizing his twos and threes on Monday as well (more on that below). The other students in that group were very close to full understanding before they came to me, and they'd achieved mastery by the end of our session.
The second group was more challenging, because one student speaks Spanish exclusively (my own Spanish is limited and rusty, but the goblins and I start back with our Spanish tutor this week so I'll be improving), and another student is reported to have an IQ of 70 and to be clinically retarded. That one is mysterious to me, because while he did certainly need remediation in topics well below fifth-grade level (we were using manipulatives to illustrate the concept of multiplication), he picked it up easily and is also thoroughly bilingual. He's apparently about to turn 13, but it's difficult to tell whether there really is an intelligence deficit or some kind of learning disorder (or disorders) muddying the waters.
On Wednesday I arrived with my own manipulatives and ready to pick up where we'd left off, but to my dismay we were moving on to long division (!). Their teacher has to move the rest of the class along, of course, and because we're working during class time we have to at least attempt to keep up. After the teacher's lesson, my first group of students made their way back to my table, and sat down not with the excited and proud faces they'd left me with on Monday, but with expressions of deep frustration and self-doubt. Still, even though we'd skipped some very important steps and they'd been terribly discouraged by trying to follow along with the lesson, they did get it working after we discussed it some as a group, and by the end of our time three ten year-old boys asked me if they could skip lunch and do more math with me. I'm taking that as a sign of success. I tried to make it clear to them that far from being dumb, they'd essentially skipped two grade levels (without mentioning the grade levels themselves) in 48 hours and still managed to succeed. I've also arranged to set up an after-school program two days a week so I can work with these students on the topics they really need to be working on so that they can follow the class lessons without frustration. The teacher asked if I could teach reading as well, and my groups are small so it's actually starting to feel a bit like homeschooling after all.
I mentioned earlier that Ozzy's working on multiplication himself. Singapore Math teaches operations and their inverses together, so the unit he's just finished in 2A covered the introduction to the concepts of both multiplication and division. When we're discussing a new topic, I always make the first few lessons a leaderless collaborative effort, the next ones a student-led collaboration, and the final ones I let him work through on his own with the option to ask questions. When he's going to be working on his own, I flip through what he'll be doing to make sure he'll be comfortable with it, but on Tuesday I somehow missed that the last two pages I'd set out for him were division, which we hadn't discussed at all yet. I'd gone down the hall to brush Sofie's hair, and when I came back five minutes later he was done, having apparently fully understood what the one-word instruction ("Divide:" followed by a list of equations) ought to mean and completed both pages without error. We've since discussed and played with division just to be thorough, but that was a fun moment.
Ozzy does have a tendency to despair if he doesn't fully assimilate a concept the first time he hears it, and working on difficult problems from multiple angles feels like cheating to him, so last night we watched the NOVA documentary on the process of proving Fermat's Last Theorem. Seeing that the world's top mathematicians struggled with that one problem for centuries, and that solving it absolutely required thinking about it from many different angles and using different approaches, seemed to help Ozzy relax and believe me that math is not a race and that part of the joy of math is the struggle and the insights that are developed during that struggle. It was also very interesting for me, because one of the classes I'm taking this quarter introduces proof techniques, which I could recognize in an extremely primitive way in the work in the film.
In my last weekly review, I mentioned that we'll begin MCT's Grammar Island in the next month or so, and I expect to transition into it's being our primary grammar curriculum instead of First Language Lessons. I had already ordered FLL3 when I discovered MCT, and from what I've read the combination of the two is rather ideal for the first level or two of MCT so that worked out fine. I haven't ordered MCT yet, but FLL arrived last week and confirmed something else I'd read about FLL: that FLL2 is unnecessary for most students. Oscar's over halfway through, and we'd been taking the lessons two or three at a time because they were fairly repetitive and moved forward in rather small increments, which might be very good for a student for whom language or memorization are not major strengths. FLL2 repeats everything covered in FLL1, and the new material in FLL2 is repeated in FLL3, so unless a student needs the repetition or some extra time to mature, it's very feasible to go from FLL1 directly into FLL3, which is what I plan to do with Sofie. In any case, Ozzy's started on FLL3 now and is doing very well with it.
The second group was more challenging, because one student speaks Spanish exclusively (my own Spanish is limited and rusty, but the goblins and I start back with our Spanish tutor this week so I'll be improving), and another student is reported to have an IQ of 70 and to be clinically retarded. That one is mysterious to me, because while he did certainly need remediation in topics well below fifth-grade level (we were using manipulatives to illustrate the concept of multiplication), he picked it up easily and is also thoroughly bilingual. He's apparently about to turn 13, but it's difficult to tell whether there really is an intelligence deficit or some kind of learning disorder (or disorders) muddying the waters.
On Wednesday I arrived with my own manipulatives and ready to pick up where we'd left off, but to my dismay we were moving on to long division (!). Their teacher has to move the rest of the class along, of course, and because we're working during class time we have to at least attempt to keep up. After the teacher's lesson, my first group of students made their way back to my table, and sat down not with the excited and proud faces they'd left me with on Monday, but with expressions of deep frustration and self-doubt. Still, even though we'd skipped some very important steps and they'd been terribly discouraged by trying to follow along with the lesson, they did get it working after we discussed it some as a group, and by the end of our time three ten year-old boys asked me if they could skip lunch and do more math with me. I'm taking that as a sign of success. I tried to make it clear to them that far from being dumb, they'd essentially skipped two grade levels (without mentioning the grade levels themselves) in 48 hours and still managed to succeed. I've also arranged to set up an after-school program two days a week so I can work with these students on the topics they really need to be working on so that they can follow the class lessons without frustration. The teacher asked if I could teach reading as well, and my groups are small so it's actually starting to feel a bit like homeschooling after all.
I mentioned earlier that Ozzy's working on multiplication himself. Singapore Math teaches operations and their inverses together, so the unit he's just finished in 2A covered the introduction to the concepts of both multiplication and division. When we're discussing a new topic, I always make the first few lessons a leaderless collaborative effort, the next ones a student-led collaboration, and the final ones I let him work through on his own with the option to ask questions. When he's going to be working on his own, I flip through what he'll be doing to make sure he'll be comfortable with it, but on Tuesday I somehow missed that the last two pages I'd set out for him were division, which we hadn't discussed at all yet. I'd gone down the hall to brush Sofie's hair, and when I came back five minutes later he was done, having apparently fully understood what the one-word instruction ("Divide:" followed by a list of equations) ought to mean and completed both pages without error. We've since discussed and played with division just to be thorough, but that was a fun moment.
Ozzy does have a tendency to despair if he doesn't fully assimilate a concept the first time he hears it, and working on difficult problems from multiple angles feels like cheating to him, so last night we watched the NOVA documentary on the process of proving Fermat's Last Theorem. Seeing that the world's top mathematicians struggled with that one problem for centuries, and that solving it absolutely required thinking about it from many different angles and using different approaches, seemed to help Ozzy relax and believe me that math is not a race and that part of the joy of math is the struggle and the insights that are developed during that struggle. It was also very interesting for me, because one of the classes I'm taking this quarter introduces proof techniques, which I could recognize in an extremely primitive way in the work in the film.
In my last weekly review, I mentioned that we'll begin MCT's Grammar Island in the next month or so, and I expect to transition into it's being our primary grammar curriculum instead of First Language Lessons. I had already ordered FLL3 when I discovered MCT, and from what I've read the combination of the two is rather ideal for the first level or two of MCT so that worked out fine. I haven't ordered MCT yet, but FLL arrived last week and confirmed something else I'd read about FLL: that FLL2 is unnecessary for most students. Oscar's over halfway through, and we'd been taking the lessons two or three at a time because they were fairly repetitive and moved forward in rather small increments, which might be very good for a student for whom language or memorization are not major strengths. FLL2 repeats everything covered in FLL1, and the new material in FLL2 is repeated in FLL3, so unless a student needs the repetition or some extra time to mature, it's very feasible to go from FLL1 directly into FLL3, which is what I plan to do with Sofie. In any case, Ozzy's started on FLL3 now and is doing very well with it.
