Accelerating without a net.

Sushi is our traditional reward for finishing a math book.

On Thursday, Alex finished MEP 4a, which is theoretically the first half of fourth grade math. I looked ahead in math to see what our likely sequence might be. On the pre-algebra pretest at the Art of Problem Solving website, the only things she can’t do now are multidigit divisors, operations with decimals, and negative numbers. Allowing for plenty of practice, she could realistically finish the elementary math sequence in another year. Which would put us on pace to start pre-algebra somewhere around her ninth birthday.

That scares me.

I am grateful that homeschooling allows us to proceed at Alex’s own pace. I am glad that we can calibrate her math work based on our own observations, without having to justify our case to an educational bureaucracy. And yet it’s also scary to be accelerating without a net. What if we’re missing something?

What if we’re self-deluded?

After all, one of the most common tropes in modern American parenting is the parent who overestimates her kid’s talent. I’ll admit that I’ve seen things written by other parents that have made me cringe. So it’s uncomfortable for me to talk about giftedness or acceleration; I vividly remember the scornful condescension with which an anonymous commenter once explained to me that Alex, while “cute” and “obviously well-exposed,” was certainly nothing unusual.

In general, I’m a fan of a “deeper, not just faster” approach to math; rather than race Alex quickly through the levels of a standard curriculum, I’ve sought out the most challenging programs I can find. I’ve been planning to run her through the majority of MEP and Beast Academy, so that she’s exposed to different teaching strategies, emphases, and enrichment topics. I’ve looked to add in fun enrichment and have contemplated substituting logic for math one day a week. And even though we’re doubling up on curricula, I have avoided compacting either program very much. After our experience with Beast Academy 3a-c indicated that she does fine with less intensive practice, I did approach MEP 4a with greater willingness to eliminate problems – but it wasn’t until near the very end that I dared to eliminate a few whole lessons.

Part of what’s been in the back of my mind, through all of that, is discomfort with the whole idea that she might hit algebra at ten or eleven years old. I’ve found myself assuming that “slowing her down” is inherently a good idea, without looking at that too closely. I haven’t, after all, wanted to be “one of THOSE parents.” Really, when it comes down to it, I’ve been afraid to accelerate in any significant way. It feels safer to have her be no more than a year or so “ahead.” It’s scary to be her parent and her teacher, making the call about sending her flying out there without the “net” of some official validation.

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18 Responses to Accelerating without a net.

1. Kim says:

Amusingly, my anti-spam word is “chill”. Lol. Not that that’s my advice to you, exactly. It is scary to think of doing algebra with a child so young. BUT, even if you have doubts of your own judgement [which is normal, even if I tend to think they are unnecessary here], I think A will let you know if it’s too much too soon, and then you can back off and find something else.
I love all the enrichment things you do. L and I are really enjoying the Perilously Perfect book, and I look forward to the Number Devil. We also have Life of Fred as a fun supplement.
Have you chosen a logic supplement? I’ve looked at a few, but most of the ones I see mentioned on TWTM are religiously-based.

2. Kyndra says:

It’s a tough call. Currently F is running about a year ahead and Su is almost two years ahead. I keep thinking that at some point we’ll run up against concepts that they aren’t ready for and that will slow us down. Until then I am trying to make sure that we are going deep and wide (adding extra subjects) and figuring that if they do run way ahead they will have plenty of time for internships and college prerequisites as teens…K

3. jw says:

I taught pre-Algebra in a gifted school to Grade 5 students. Many of them found it trivially easy. I don’t know if that helps or not, but there it is.

I think, in terms of what should be taught when outside of a mandated curriculum, you have to be led by the child. If you’re not missing anything and she’s ready, why not move forward? If you go too far, she’ll let you know. She’ll balk or something. As long as it’s not a case of doing ALL the math and ignoring, say, writing.

I get your concerns about pushing her or about her going through things too fast – I have dealt with parents who do that – but there’s a very real difference between pushing and challenging.

Also, I’ve been working in a school full of gifted kids for the last 8 years. Knowing Alex, admittedly only through the pictures you paint of her, I think she’d fit right in, for whatever that’s worth.

4. Ian Osmond says:

Seems to me pretty likely that, if you find out that you overshot, you can go back and pick up anything you missed. I don’t know that for sure — I’ve not studied educational theory or child development in any sort of depth, but it seems to me that that would make sense.

5. Something you might find helpful or at least an interesting read — back when Molly was nine, I asked for advice from my friends who had been mathematically gifted children. My sister re-posted to her LJ; she went to Harvey Mudd for college, so she knew a lot of hard-core math geeks. Anyway, I got a whole lot of advice, some of it really helpful: http://naomikritzer.livejournal.com/226731.html

I think I want to have my coffee before I tackle this in more detail, but I’ll note before I go that my husband was perpetually worried when Molly was younger that she was going too fast and not learning things WELL. That she was missing things, etc. She has on occasion run up against something she missed; she figures it out and keeps going. It’s not a big deal.

Something else I’ll add is that as a lot of people noted in my LJ, the high school sequence is not set in stone when you’re doing things independently. Algebra is the foundation for a lot of stuff, but if you want to keep her from careening toward Calculus you can spend a good long time studying statistics, for instance.

6. Mary says:

I did notice once before that you reassured your readers that you didn’t want Alex to hit algebra at some younger age (8? 10?) as if it was assumed to be a bad thing in almost all cases. My husband also noticed that I think: he was institutionally schooled, but his school teacher mother privately accelerated him in maths for some years. It was a neutral-to-good thing for him. I wish I’d been exposed to high school mathematics a few years earlier myself.

Also frankly, my entirely anecdotal experience is that I could have used a lot more experience before university of maths classes moving too fast for me: this is an important skill. Unless Alex is very gifted in maths (and I can’t tell! she might be!), say at maybe the 1 in 10000 or 1 in 100000 level of ability, if she pursues mathematics education for a long time at some point the classes will move awfully quickly from her perspective! This transition was very abrupt for me and I had no coping skills ready for it. But in general this is very in keeping with your homeschool philosophy: Alex experiences academic challenges throughout her schooling so that she will be ready for the challenges of undergraduate and postgraduate level work which is where many students have to learn to deal with challenges all of a sudden. As a reader I admire your work in this direction and I’d be very surprised if you’re not doing Alex’s maths education very well in both depth and speed.

7. Jinian says:

I did pre-algebra for the first time at 9-10. Granted, after that they didn’t know what to do with me for a while, so I didn’t get algebra until 13 or so. My math skills weren’t harmed by studying it early, though my later homework-apathy may have had something to do with the holding pattern.

I definitely agree that the Canonical Math Sequence is an artifact of how schools teach math and not a reflection of the interesting parts of math or what makes sense together. You can do a LOT of geometry and mathematical logic without ever needing calculus, and the way we teach trig without much geometric reasoning is IMHO counterproductive for most people. There may not be curricula available for all that stuff, but Alex seems very amenable to setting up her own projects now so I bet she’d be all over making lesson plans for herself in a few years.

8. Nolly says:

I think the pace of public-school math education has a lot to do with overcoming enculturated math-phobia, too. Without that obstacle, a lot of kids — not all, but a lot — could move faster.

9. Okay, a couple more thoughts now that I’m caffeinated.

1. On multiple occasions I’ve let my fear of being One of Those Parents (or, really, my fear of being perceived as One of Those Parents) get in the way of doing the right thing for Molly. This was a mistake. You’re a homeschooler; you don’t have to convince anyone else that Alex is ready for pre-algebra. If she’s ready for it, go ahead and start her.

2. Looking at that pre-algebra test, I’m kind of surprised at how little it covers. I would have thought that mixed fractions (and multiplication and division with fractions) would be something you would learn before pre-algebra — apparently that’s something the pre-algebra book covers. There’s also nothing in there about exponents, square roots (not that kids need to be able to calculate them by hand, but the concept of squares and square roots is important), order of operations, and basic geometry of the “here’s how to find the area of a triangle” type. Given all that’s not in here, I’m REALLY surprised that their pre-algebra course is designed to be a one-year sort of thing.

I can’t actually remember what all Molly’s pre-algebra class last year covered, but it included a lot of basic algebra — and you’d absolutely need the fractions, squares/square roots, and order of operations stuff in your head before you started doing the basic algebra stuff.

I should note that I’m not super familiar with the AoPS curriculum, other than looking at it a little bit online in the past. I do remember liking what they said about their approach.

3. Given all the pieces that seem to be missing from this test — AoPS pre-algebra just doesn’t look to me like it would be frighteningly accelerated. So don’t worry too much.

4. Have you used Hands-On Equations with Alex at all? This is essentially basic algebra with manipulatives for early elementary schoolers. Molly looooooooved Hands-On Equations.

5. Mary’s comment about the speed of math classes — the thing that Molly ran up against a lot was that math classes at every level ran too slowly. Fourth grade was probably the most frustrating because her school was deliberately holding her back (she was ready for pre-algebra; they didn’t want her to take it because they didn’t have an easy way to teach her algebra in 5th grade.) In 5th grade, she had pre-algebra which covered new material but moved way too slowly for her. Her program at the university, by contrast, moves absurdly quickly — they cover Algebra I in one semester, Algebra II in the next semester, and complete the high school mathematics sequence in two years. For some kids this would be nightmarish (it would have been DISASTROUS for me) but for the kids who do UMTYMP it’s generally fantastic.

My kids just got home; maybe I’ll have more thoughts later.

10. Molly, after reading the Beast Academy / Perilous Math post and this one: “I think she should go ahead and accelerate Alex.” She also thinks you should relocate to Minnesota, so that Alex could be in UMTYMP. (She was looking at the calculations Alex made for the laser problem and — on hearing that Alex was eight — said, “Wow, she’s really good at math. She should try to get into UMTYMP when she’s old enough!” When I noted that she lived in Maryland, Molly said, “oh, well, they have plenty of time to move.”)

Molly thinks she started long division in 2nd grade, and learned multi-digit divisors in third. She learned fractions in 2nd grade, but not at school. Decimals in 3rd (she thinks, she’s not sure exactly).

Then she changed schools and in 4th grade she did mostly stuff she already knew. It was Singapore Math 5a and 5b and was actually review for most of her peers, as well. (5b was better in that respect than 5a.) (OH and they initially put her in the class that was doing 4b/5a, despite having been told that she was really good at math and working well ahead of grade level because no one ever believes me when I tell them that Molly’s really good at math. Molly says that they probably get a lot of parents saying, “she’s really good at math!” when the kid is average at math for this particular school. “Sort of the way Leni [her Kindergarten teacher] assumed that when you said I could read, she thought I was reading ‘See Spot Run’ or that I’d memorized a book.”)

Molly suggests that for enrichment you be sure to get her “Math for Smarty Pants,” because it covers all sorts of interesting things that are not covered in normal math programs (like perfect numbers).

Oh, and also, Molly says that if ALEX thinks she’s ready for something, then she probably is.

11. tinderbox says:

Thanks for your comments, everyone. I’ll probably be reading through them several more times. I am particularly grateful to Molly for the kids’-eye view.

It’s helpful just to get some data outside my sample of 1. There are two things that have been worrying me, which are both things that “they” say about math acceleration: (1) that accelerated kids are just memorizing procedures without true understanding, and (2) that the ability to do algebra takes developmental maturation that is more or less independent of the speed of prior math acquisition. So it’s helpful to hear that j.w.’s students routinely learned pre-algebra in 5th grade, and that Molly, Jinian, and others studied it early with no apparent ill effects.

Naomi, I had the same feeling of bafflement at how easy the AoPS pre-algebra pretest is, because it is not, seriously not known as an easy curriculum series. Of the things you mention, Alex has already had order of operations and squares and square roots, and we’ll be doing basic geometry of triangles at the end of Beast Academy 3d. I’m sure they’re assuming pre-algebra students have already had it. (Maybe someone out there whose kid is doing AoPS pre-algebra can weigh in?)

12. Molly says:

Not AoPS, but Mia is doing EPGY Pre-Algebra right now. It’s supposed to be “Honors Pre-Algebra” but they do assume knowledge of basic geometry of triangles. They review it briefly. Right now she’s learning sine, cosine, and tangent, as well as applications.

I looked at the pre-algebra AoPS test and it looks like it would be a test to get into 5th grade math for EPGY. Mia took that last year so I’m familiar with it. Although they did teach long division (reflected here), a great emphasis was placed on multiplying and dividing fractions, adding and subtracting with different denominators, etc. To my surprise, it had quite a lot on set theory as well. EPGY has 5th grade math, 6th grade math, and then Pre-Algebra.

I note this is a test for “Pre-Algebra 1.” This may not apply, but in my younger kid’s school, there are technically three grades of Pre-Algebra. “Pre-Algebra 1″ is code for 6th grade math. So a child starting Pre-Algebra 1 in 6th grade and not skipping would enter Algebra in 9th grade.

13. tinderbox says:

Molly, I think it’s right that it is meant to be 6th grade math. They’re planning to carry Beast Academy up to 5th grade, with the assumption that kids will then be prepared for pre-algebra. “Pre-algebra 1″ and “Pre-Algebra 2″ are semester courses rather than year courses, so I guess the assumption is algebra in 7th grade. I haven’t looked at EPGY at all.

Interestingly, our non-Beast curriculum, MEP, has a 6th year. I am not sure how it compares.

14. Kiya Nicoll says:

I did self-paced maths for much of elementary school, while doing institutional schooling for the rest of it (as I believe you might recall from my past ramblings thereupon). By the time I was ten, I had done pre-algebra, algebra, and fragmentary bits of Algebra II (as it was called there/then; I don’t know what is the current thing), and then we moved, and the different institutional system shit kittens at us because they could not imagine that I could have done such a thing and thus they put me in sixth grade math. (The advanced math course in Montgomery County would have algebra in 8th grade and algebra II in tenth, and a single year of advancement was not common but certainly not unheard of.)

My experience would be that running into other people’s assumptions about how it all works and what a child is capable of can be very hard to manage. Further, one thing that tripped me up was that (once we got all that sorted out) I ran out of math to learn before I had enough high school maths credits for graduation in the state of Maryland, which was inconvenient for me and kind of embarassing for the administration. I do well at math where I have a solid grasp of applications and what it’s good for (which includes ‘this is fun to play with’ as an application), but taking more advanced math without related science or other material-world groundings wasn’t going to work well for me; it just plain wouldn’t stay in my head. So those are things I would flag as potential things to think about as part of your future planning.

(Also, I have been taught matrix operations three times, all of them badly and in ways that did not take, and thus I run into a wall made out of large square brackets somewhere around linear algebra.)

15. ailbhe says:

I am utterly bewildered by the idea of waiting a long time to introduce algebra. It’s tremendous fun, comforting and reliable, and if someone finds it easy to grasp they can do it. L intuited a lot of the basics before she could do long division even. If she doesn’t like it she can do something else instead.

Um apparently I have strong feelings on this.

16. jengod says:

Don’t be scared of algebra. Heck, don’t be scared of calculus. Americans, as a general rule, are terribly easy on themselves when it comes to math. Students in India, Singapore, Hong Kong, Korea and Japan do what we consider to be “college-level” math in high school. For that matter, in other countries, physics isn’t considered the unconquerable beast that it’s thought to be state-side. Your kid can do it if you guys decide that’s a good idea, and you should have no qualms about it whatsoever.

You don’t owe anybody an explanation or an apology. Go for it.

17. arti says:

I would hold off algebra for the same reason I would hold off calculus for as long as I can, not because they are difficult but because they are easy.
I did a lot at algebra and geometry in school. A lot of it consisted of minimizing or maximizing functions using what looked like ad hoc methods, finding areas and volumes using different formulas etc. And then we got into calculus and we learned how to easily find extremums of arbitrary functions and complicated areas and volumes and so on. For me the realization was so exhilarating that I fell in love with math and I have been in love ever since. I don’t think the experience would have been as joyful if I wouldn’t have previously spent so much time with algebra and geometry.
Just like calculus shows one how to solve large groups of algebra problems, algebra also shows a canonical way to solve large groups of arithmetical problems. Algebra’s beauty cannot really be appreciated if one hasn’t spent enough time using what seem like ad hoc methods to solve arithmetical problems.
Instead of doing the Pre-Algebra pretest to see if the child is ready to start algebra I would make sure that she demonstrates great knowledge of arithmetics by being able to do lots of different problems. The best collections of such problems that I know of is “Ray’s New intellectual arithmetic”. You can download this book for free from Google books. Yes, it is a very very old book and there are much better ways to solve the problems arithmetically than those described in the book, but the problems themselves are great. If your child can do those she will love algebra.

18. Kim says:

If she is ready for it or not when the time comes, I am sure you will know it, and adjust if necessary. This is the beauty of homeschooling – had she been in a traditional public classroom, she would have been held back from her potential, because she would not have gotten a personalized education. “Gifted” or not, I think many kids could be “accelerated” (which is only when you are comparing to someone else…) in a homeschool environment, because each child has his/her own strengths.