I’m almost a week late writing Alex’s quarterly progress report. Good thing the due dates are just self-imposed, huh? We’ve had a great summer, with plenty of time for new friends, hanging out, and playing, but also some good academic progress. It’s nice to feel ahead of the game going in to fall – we’re moving in October, so we’ll be losing a lot of our time to house projects and packing.
Language Arts: Alex discovered comic strips this summer, and spent most of her free reading time with Calvin and Hobbes collections. Those things actually have a high vocabulary level and require a lot of ability to make inferences, so it’s not a step down as far as reading practice is concerned. She’s also been happily working her way through the American Girls historical fiction series and various middle-grade novels. I continue to get piles of picture books that “go along” with our Five in a Row book of the week, but it’s rare that I read them aloud anymore – Alex reads them on her own, and then shares the science or history facts she’s picked up. She continues to test at a mid-fifth-grade reading level.
As I reported recently, Alex’s writing has really improved this summer. She’s started to do some free writing of her own accord, and is using my scribing services less during math lessons. We’re on track with where I’d like to be, but I think she’d still struggle with the quantity of writing required if she went to public school.
This summer, Alex came close to finishing All About Spelling Level 1. We’re on the second-to-last lesson and have begun to hit some challenging concepts. Alex has mastered concepts such as choosing k vs. ck at the end of a word and c vs. k at the beginning of a word. She can spell simple short vowel words reliably by breaking them down into component sounds. Her spelling skills don’t necessarily transfer to free writing, but I think that’s pretty normal at this stage.
Math: Alex finished MEP 2a this summer and started 2b. Most of her time this summer went towards developing a really thorough understanding of two-digit addition and subtraction. There were some interludes with measurement and easy geometry, which provided a welcome break.
It’s been interesting to me to see that her curriculum hasn’t presented the standard algorithms for addition and subtraction yet – what we used to call “borrowing” and “carrying,” but is now called “regrouping.” That doesn’t come up until 3b. Instead, the main focus is on mental math techniques. She adds 38 + 46 left-to-right by figuring 38 + 40 = 78, + (2 + 4) = 84. It sounds cumbersome, but she’s pretty quick. I think she’ll wind up with a better conceptual foundation, and a better ability to calculate on the fly, than I had as a child.
I’m not sure what our math future will be beyond MEP 2b. Looking ahead, it might be wise to slow down our rapid forward progress and go deeper and wider instead. Art of Problem Solving, a company which produces advanced math texts for high-performing middle and high school students, is putting together a new curriculum called “Beast Academy” for grades 2-5. The textbooks are in a comic book format with a strong narrative, and people who have reviewed a sample chapter say that the lessons go very deeply into higher-order reasoning. If we integrated Beast Academy and MEP after we finish 2b, that could keep the challenge level up while slowing us down so she doesn’t hit algebra at age 10. And I think Alex would respond very well to the story format.
Five in a Row: This quarter we studied eight FIAR books, plus one extra study that I put together on Plains Indians and buffalo. We’re actually starting to wind down Five in a Row; I planned out our books for the fall quarter and realized that after Christmas we’d only have a few books left in Volumes 1-3. I do have Volume 4, which includes more extensive studies meant to be completed over two weeks, so we’ll start that this winter. But I’ve been feeling lately that FIAR occupies a less central place in our day than it used to. I don’t know whether that’s because Alex is starting to outgrow it, or if it’s just because after 30-some books I’ve lost some of my excitement about all-out planning. We’ll see how the fall goes, and then the jump to Volume 4.
History: I thought we would finish up Story of the World, Ancient Times this summer and be ready to start the Medieval Era in the fall. Instead, with one thing and another, we just haven’t done much history this summer. We’ve just hit the rise of Julius Caesar, after taking a break from the early history of Rome to tour ancient India and China. We’ve got another six chapters or so of Rome to go, and that will finish the book. Alex continues to love history. A highlight of the summer was her Roman story.
Latin: Latin is going great. We’re about halfway through Song School Latin and going strong, plus both kids have learned their amo amas amat thanks to a silly YouTube video. We may be finished with Song School by Christmas, so I’ve been shopping around to find where to go next.
?
My father taught me to do basic arithmetic in a style very similar to what Alex is learning when I was about the same age as she is. I know it makes Andrew look at me oddly when I start doing math in my head but my basic calculations are faster and more reliable than his.
I’d approach the problem of 38+46 a little differently than what you described-I’d round 38 up to 40 and then calculate 40+46=86, then I’d subtract 2 and get 84.
And I would do that math problem in yet a different way: 38 + 46 = 40 + 44 = 84.
I do two-digit multiplication in stages (usually First Inner Outer Last), too, and generally out loud to keep track. It amuses the heck out of my coworkers when I start chanting 38 x 46 = 1200 + 320 is 1520 + 180 is 1620 + 80 is 1700 + 48 is 1748.
I really enjoy your Tinderbox posts.
JanetM
My friend Laura Laing recently published a book called Math for Grownups. One of the things she points out is that in any random group of adults, you might have five or six different ways of approaching that problem, all of which work… but probably no one solving the problem mentally says “8 + 6 = 14, put down the four and carry the one, 1 + 3+ 4 = 8, 84.”
The method I use – and Alex also uses it too – is to say “30 + 40 = 70, 8 + 6 = 14, seventy-fourteen, which is 84.”
Coming up this term, Alex will be asked to memorize the times tables. That should be interesting. I remember how much I hated the brute memorization techniques we were given for learning them.
Like Erika, my dad taught me to do mental math that way when I was a kid. I wonder if that approach was commonly taught in the pre-calculator age? I think the visuals in my math texts helped, too — cubes, lines (stacks of ten cubes), squares (ten stacks of ten cubes) for 1s, 10s, 100s. I remember imagining people living in adjacent apartments with gadgets in the walls, so that when Mr. 1s needed to borrow some, Mr. 10s would pass a 10-stick through and it would get chopped up into cubes for him.
My mother didn’t learn her times tables and it basically put an end to maths education for her. She eventually became a primary school teacher and was worried about her maths only going up to about Year 3 or 4, but happened on some tapes that put the tables to music, so she learned them from the tapes and used them with her classroom for years too. Eventually she became known as the teacher who could extend Year 6, which was very awkward, as she can with puzzles and such but she can’t do standard Year 7 problems. (In NSW, high school starts at Year 7. Teachers are trained for K–6 or 7–12, so 6 is a common boundary for “unable to teach”. It’s a big change in maths in particular too, with both algebra and curve graphing introduced in Year 7.)
I found it quite useful to look for patterns in the times tables and I think that helped me. For example, if you construct a 12×12 square and write all the answers in it—first row reads 1,2,3…, second 2,4,6…, third 3,6,9….—it helps show visually that multiplication is commutative and the diagonal (which will be square numbers) is interesting. I expect the music mechanism would have worked as well.
Fascinating – when I try to break down how I do addition in my head, that method you described for 38+46 is pretty close to what I come up with. It’s almost a form of estimating, then putting the precision back in with the small stuff before you give a final answer, or that’s how it feels to me, and I wish someone had told me it was acceptable as a way to answer math problems when I was a kid.
As a side note, 10-11 is a really good year for algebra. I found it a lot easier then than when I had to relearn it because they made me wait to take the later classes.