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2013 August 8

Hard Math for Elementary School

Filed under: Uncategorized — gasstationwithoutpumps @ 20:34
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Hard Math for Elementary School by Glenn Ellison: (ISBN 9781489507174) looks like a book I could have used with my son about 8 years ago (too bad it was just published a couple of months ago).

The premise is simple: it is a math enrichment textbook, intended to take the topics of elementary school math deeper for gifted students.

The presentation is good, but the students will have to be reading at the 4th grade level as well as doing math at that level to get much out of the book.  This is not a flashy book with lots of illustrations and stories—it is just cool math, presented as cool math.

Disclaimer: I don’t have a copy of the book, and I haven’t read much of it. I used Amazon’s “Look Inside” feature to look at the table of contents and a couple of pages, and saw such gems as calculation tricks for computing some squares (like 552=3025) quickly.  (The trick is based on (x+y)(x-y)=x^2-y^2, but the author wisely avoids algebraic notation.)

Reviews from people who have looked at it in more detail can be found at and

Glenn also has a book for middle school students: Hard Math for Middle School

2012 July 21

National Parenting Gifted Children Week

I just found out that last week was “National Parenting Gifted Children Week“—another one of those designated weeks that no one hears about (like National Folic Acid Awareness Week and National Engineers Week).

I found out about the week (too late to do anything, not that there is anything offered to do) from a book publisher that publishes a lot of support books for parents and teachers (which some people find useful, though I never have): Prufrock Press – blog – National Parenting Gifted Children Week.

I suppose that getting something designated as “National x Week” makes the people who argue for it feel like they’ve accomplished something, though it has never been clear to me exactly what.  Since there are so many “National x Weeks” it buys you no publicity, unless you have events scheduled all over the country simultaneously (like with National Bike Month and Bike-to-Work Week).  Incidentally, that event started as a number of local celebrations that gradually coalesced into a national movement that the League of American Bicyclists and other existing bike advocacy organizations joined—the locally run events are still the mainstay of the activity, though there are multiple efforts to try to coordinate them (not just LAB, but also

National Engineers Week is trying to get engineering schools and faculty to put on events, but without a lot of success, because they chose a poor week for faculty or student involvement entirely for political reasons.  (Bike-to-Work week, in contrast, was chosen as a compromise between the various “beginning of the cycling season” dates around the country.)

National Parenting Gifted Children Week has no special activities (other than a blog tour that was not mentioned on any of the several parents-of-gifted-kids mailing lists that I read, so probably only reached the people who already read those blogs).  Unfortunately, this sort of feeble, invisible PR effort seems to be typical of organizations like NAGC and SENG. I’m sure that they mean well, but I can’t see enough positive results from their efforts to want to join the organizations.  There are 1000s of well-meaning organizations that want my time and money (and dozens who send me requests).  I’d rather support organizations that seem to be efficient and effective (like Second Harvest Food Bank and Planned Parenthood), even if I have no expectation of ever having any use for their services.

2012 May 13

Advanced Placement for talent development

Filed under: home school — gasstationwithoutpumps @ 15:15
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Since we are in the middle of AP testing for this year, it is timely that Hoagies’ Gifted has just posted Dr. Joyce VanTassel-Baska’s keynote speech from August 2000: The Role of Advanced Placement in Talent Development.

In this speech to teachers of Advanced Placement courses, Dr. VanTassel-Baska argues that AP courses are critical for gifted students, as a means of differentiating instruction for them. She said things like

In the late 1970′s gifted students reported AP to be the most beneficial program taken during their high school years. This perception has not changed appreciably over the intervening decades (Kolitch & Brody, 1992).

Of course, since she gave the talk in 2000, there has been a big push to get more students into AP courses, so that many of the courses are no longer all that suited for gifted kids.  Even at that time, she pointed out

One issue that Advanced Placement teachers need to be aware of is the different levels of aptitude for a particular AP course. The range of ability in AP classes is typically very great. Even if all students were identified as gifted, the range would be as broad as in heterogeneous classes. Such differences in aptitude level require more attention to addressing individual needs. Because AP course work probes depth of understanding, it tends to reveal greater disparity in student learning. Level of aptitude may predict how much material students can handle well, how capable they are to work independently, and how strong they are conceptually with the material. Use of various forms of flexible grouping for in-class work may be an antidote to this problem. Organizing sections of AP by ability levels may also be useful in subjects where enrollments are sufficiently high.

The trend to lower the barrier to entry for AP courses over the last decade has made this problem worse. The teaching strategies that are most effective for the bottom half of a current AP course may be quite unsuitable for the kids in the top quarter, who don’t need many routine practice problems, but rather a smaller number of more challenging questions that stretch their minds.

While AP courses may still be the best option available to most gifted high school students, many need to look elsewhere for teaching at their level. Community college courses sometimes provide this, though that varies enormously, as many community college courses are intended to remediate inadequate high school preparation for college, and may be taught in an even less suitable way for gifted students than AP courses.

Sometimes sufficient challenge can be found by skipping prereqs—the review built into the beginning of most courses can serve as a fast-paced introduction to the material for a gifted student.  This is essentially what we did by teaching my son a course that covers AP Physics C: Mechanics, with no previous physics courses.  He had picked up almost everything from a conceptual physics course by reading various popular books about physics, and his math was strong enough that he did not need the crutch of algebra-based physics, but could jump right into calculus-based physics.  (Actually, I think that calculus-based physics is somewhat easier, if you have the math, since there are fewer formulas to memorize—a lot can be trivially rederived from the definitions of force as the derivative of momentum with respect to time and of potential energy with respect to displacement, for example.)

I think that one problem with AP classes as a primary means of teaching gifted high school students is an extension of a common problem in gifted education in lower grades: what the students need is work that is more complex and sophisticated, not more work.  Too many AP courses pile on drudgery, in the mistaken belief that this makes them more like college courses. (See my analysis in How many AP courses are too many?)  This overload of marginally useful work prevents them from taking on more independent, non-curricular projects (like science fair, debate, theater, or internships) from which they would actually learn more.

I’m not saying that I think that AP courses are a bad idea, nor that gifted students should avoid them.  There are times when an AP course is precisely what a gifted student needs and other times when it is the best available choice. Taking AP courses is one of many ways for gifted students to learn, just not always the best way.

2012 April 18

Distance learning for gifted kids

Suki Wessling, a local writer who is home-schooling her kids, recently wrote an article about distance-learning oppoturnites for gifted kids: Boutique distance learning offers variety for gifted kids – National gifted children | We have not used any of the “boutique” services she mentioned, nor, for that matter the large services like Johns Hopkins University’s Center for Talented Youth or Stanford’s Education Program for Gifted Youth.

There are several reasons we’ve been reluctant to use many on-line courses:

  • Many are quite expensive. EPGY courses are around $500 to $750, plus $50 registration and shipping fees, JHU-CTY courses are $500–$1280.  I’d want to know that the course would be a very good fit and of higher quality than a corresponding community college class (about $300) before committing to an online course.
  • Too much screen time.  My son already spends more time in front of a screen than is healthy (as do I, so I can’t chide him too much). At least with community college classes he gets the exercise of bicycling to the class (in fact, this provides so much exercise that it counts as his PE class: about 4 hours a week).
  • Difficulty in finding courses that fit his educational needs and interests.  There are undoubtedly a number of courses that would be an excellent fit for him, but it is very difficult to distinguish them from other courses that have similar descriptions but would be at the wrong pace, wrong level, or have too much busy work.

So far we have only used one on-line course provider: Art of Problem Solving.  A year ago, I posted about our experience with with their precalculus course: Good online math classes.  My son did their calculus class this year with the same instructor, and we had similarly good results.  The AoPS calculus classes are not cheap ($500 with books), but they were an excellent fit for my son. If I could be assured of as good a fit in other online courses, I would be more willing to use online providers.

This year my son has been keeping time logs for his consultant teacher in the home-school umbrella.  For the AoPS calculus class that just ended, he did almost all the weekly and challenge problems, but not quite all. We added up the total hours (class and homework) for February and March, and got 56 hours—just under 7 hours a week.  His total workload for all courses (including the cycling that counts as PE) averaged 40.75 hours a week in February, which I regard as about the right amount of time for a high school student to be spending on school.  It is certainly much larger than the 2–3 hours a day that some home schoolers regard as adequate.  The main advantage for us of home schooling is not a reduction in workload, but a spending the time on appropriate work, rather than busy work or dead time.

I think that the calculus class was a good deal higher workload than the Precalculus class last year, but we did not keep time logs then, so I may be mistaken.  My son did not take any of their lower-level classes, so I can’t comment on the workload of any of them (though we did use the intro algebra and intro geometry books some earlier, and were happy with them, which is why I was willing to give AoPS online courses a chance).

My understanding is that by the end of the AoPS calculus course well over half the students had dropped, possibly because they could not keep up with the pace or the workload.  You only get your money refunded if you drop in the first 3 weeks, so a lot of families ended up wasting the tuition money.  I’m afraid of a similar thing happening if we pick an online course that is not a good fit for our son.

He will probably do one AP practice test before taking the AP Calculus BC test next month, but that should only take about 3.5 hours.  The AP test should be a good review of the essential material of the course, but so far as I can tell, the AoPS Calculus class covers more material in greater depth than the usual AP calculus BC course or the usual first-year college calculus class.  It is definitely a calculus-for-mathematicians course, with a lot of emphasis on problem solving and rigorous foundations (like using Darboux integrals, a somewhat cleaner equivalent to Riemann integrals).  Some of the differential and integral equations they had in the last challenge set seemed difficult even for me (though I must admit that ODE was never my favorite subject, and it has been over 30 years since I last did any differential equation other than a trivial exponential decay).

The AoPS courses also cover complex numbers fairly well, something that is not always done in other precalculus and calculus classes. Another gifted high school student I know has taken calculus through multi-variable calculus at the local community college.  I was amazed to find out that he’d had almost nothing about complex numbers: not even such fundamental things as Euler’s formula: e^{i\theta} = \cos \theta + i \sin \theta.  This lack came to light during physics class, when I was deriving acceleration for something moving in a circle by taking the second derivative of R e^{i \omega t} with respect to t.  It is so much easier to work with exponential functions than trig functions that it didn’t occur to me that the community college calculus classes would not have covered it.

2012 April 15

Flexible ability grouping

Filed under: Uncategorized — gasstationwithoutpumps @ 16:52
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About a month ago Charlie Boss wrote an article in The Columbus Dispatch: Students benefiting from new measures, which talked about a school using “flexible ability grouping” successfully.  The school uses data from standardized testing to try to match up students and teachers.

The ability grouping idea is an old one: it is as lot easier to teach a group of kids if they are all at about the same level and need to learn roughly the same things.  This was the original impetus for having kids in separate grades, rather than mixing 3rd graders and high schoolers in the same classroom.  Over the years, it became popular to group kids first by age (a rough predictor of academic level) and then by ability level (a method called “tracking”).

In the 1970s, it was noticed that the grouping by “ability” often relied on very subjective or distorted estimates of ability and resulted in fairly severe discrimination by race and social class.  This resulted in a “detracking” movement that grouped kids purely by age, independent of ability.  In recent years, this movement has dominated educational policy, resulting in “mainstreaming”, where even kids with severe emotional and cognitive disabilities were thrown into the mix.

Teachers have been expected to “differentiate” instruction across an ever wider span of ability and prior achievement.  An 8th grade teacher might have in the same classroom kids ready for college-level work and those who can barely read at a 1st or 2nd grade level.

Some educational policy makers are trying to eliminate the impossible demands on the teachers by clustering the kids so that the kids in a class have more closely matched instructional needs.  They need to do this without bringing back the racial and cultural prejudices that killed off tracking.

The solution is fairly simple and fairly obvious, nicely summarized in one sentence:

Hannah Ashton also allows students to move up or down through the groups, depending on their needs and abilities.

The problem with tracking was that kids were classified once (often wrongly) and then kept in the same “track” forever.  Ability grouping (or, as I prefer to view it placement by achievement) results in kids moving from group to group as they progress.  Some kids will race through the material, changing groups often, while others will take longer to master the material.

Each subject should have independent grouping, so that a kid who struggles with math while racing through literature does not have to be held back in the English classes, and vice versa.

The Hannah Ashton school does one other thing—trying to match the teachers to the students.  Some teachers excel at teaching the slower learners, having the enormous patience and multiple teaching approaches needed to get through to them, but would flounder when faced with a class full of very bright kids who perpetually challenge the teacher’s authority and knowledge.  Similarly a teacher with the deep content knowledge and quick wits needed for the very fast students might lack the patience and pedagogic techniques needed for the slower ones.

It isn’t a matter of assigning the “best” teachers to the slowest students or the fastest students, but of trying to match the strengths of the teachers to the needs of the students.  There is no global “best”—the best teacher for my son may be the worst one for yours.

Of course, I have no idea how good a job of matching teachers to classes the school administration does, but I’m impressed that they are trying it at all.  Even if they don’t get the matching quite right, the simple act of clustering kids into groups with roughly the same needs should make the teaching much more effective for all the students.

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