Yes, basically. Tracking as currently constructed in education really reinforces “fixed” mindset. I am open to the idea that you can have accelerated options that don’t reinforce that way of thinking, but the problem is that the people with aptitude (and their parents) really like the message of “you are super special and most people couldn’t be as special as you even if they tried their hardest”. So if there’s an opportunity to spin it that way, it gets spun.
No, I definitely see that even at the upper elementary level, and a significant part of my duty with the AIG-identified kids is to persuade them that struggle is the sign of a good learner, not a bad learner. (Some of them already know this, but there’s a significant chunk that don’t).
Today I had a kid say something really interesting to me. The lesson, per request by the classroom teachers, was for me to talk with the kids about applying the grammar stuff I’m teaching to all their writing, not just to writing for “writing class.” So I did this whole lesson around the amazingness of language, the history of oral and written language, the uses of language, and how the conventions of written language were harder to learn and more arbitrary than the natural grammar of oral language, but that using the conventions is crucial to effective written communication. The upshot was that kids should follow those written conventions across all written communication (with provisos made for special conventions in certain arenas like texting), or else their communication would suffer.
One kid, very smart, raised their hand and said (paraphrased), “When strangers see my writing, they say it’s great. But when teachers see it, they say I’ve made mistakes. How come?” They were kinda of challenging me: maybe this whole “follow written conventions” thing was just a teacher hangup?
The answer is surely obvious to y’all, and I gently explained that the strangers were showing good manners by complimenting a child’s writing, that correcting the written conventions would be super-rude. Teachers are in the role where it’s not rude to correct; rather, it’s central to the job.
The kid was very taken aback, and I’m pretty sure I know why: I think they’ve received the message that they’re intrinsically a brilliant writer, and that means that they can safely ignore the advice of teachers who clearly aren’t brilliant writers (or else they’d be published authors instead of elementary teachers). Convincing kids like this one that their natural talent must be coupled with a humble willingness to work, to accept constructive criticism, and to hone skills–this is a key part of my job.
And the even more bitter corollary: if you don’t, you will fall behind, and those kids who work harder will actually end up more skilled than you. The high-aptitude kid who coasts ends up behind, not just on par. They then feel ashamed (because they were “supposed” to be naturally better) and the shame turns to resentment and hopelessness.
I think this is exacerbated by the fact that a certain type of gifted “edge” is one of the first thing to go with age and neglect: that rapid, fluid memory that, IME, the majority of “highly gifted” kids have. It’s learning every country and it’s capital almost effortlessly, remembering every word of 1000 Wikipedia articles. It even looks like good writing, because you remember a huge library of stock phrases that elevate writing. In math, it’s remembering all your factors and squares. That talent evaporates very quickly after 20 or so, if you don’t use it every day (and some of it goes, regardless.).
That fluid, easy memory, that sponge-ness, is not the entirety of “highly intelligent”, but it’s the fun part, the part that startles peers and amazes teachers. When it starts to go, it’s easy to feel like you’re losing something essential to you–and that’s when you need to have already developed something else meaningful to take its place.
Yeah, I see all the Facebook memes from adult “gifted kids” who feel like abject failures as adults and who, worse, blame their gifted program for that feeling. They’re a warning story. The younger I can convince my students that passionate work is both fulfilling and necessary, the better.
I had a kid some time back who was freaking out because he couldn’t solve a tricky math problem I gave him (something like “Make $4.10 out of 30 nickels and quarters”). I showed him a non-algebraic way to solve it* and suggested he try it. “But that’s tedious!” he objected, and continued to refuse to attempt the problem until I had to go on to help other folks.
If nobody can reach that kid, things are gonna get real hard.
(in case anyone cares, here’s the tedious method, taught through some guiding questions, and which other kids jumped at the chance to use):
Note that the $0.10 of $4.10 must be two nickels. Now you have to make $4.00 using 28 coins.
Note that the remaining nickels must show up in groups of 5, or else you won’t be able to add them to quarters to get a whole dollar amount. Now you’ve got 5 possibilities to check:
5 nickels, 23 quarters
10 nickels, 18 quarters
15 nickels, 13 quarters
20 nickels, 8 quarters
25 nickels, 3 quarters.
Note that 16 quarters equals $4.00, and use that to rule out two possibilities.
Check the other three.
There’s a difference between the “gifted” labeling, which encourages the above mindset by treating capability at education as a single quantity (one that is given out randomly as a “gift” by God or genetics), and putting people in appropriate classes for different subjects.
Putting someone who is 5 years ahead of his peers in reading/writing ability into more and more challenging math classes until that person realizes that no one is the best at everything and there’s always a bigger fish in the sea can be valuable precisely for teaching that lesson and avoiding the “smart kids never learning work ethic” problem. This is one of the most crucial functions of “super elite” programs like the triple-accelerated track and the magnet schools that Virginia’s progressocrats are desperately trying to get rid of.
Out of curiosity I solved it myself without using algebra before looking, but turns out I used a different method anyway.
I agree with this. The point to me is to put kids in appropriate classes so they’ll be challenged but not overwhelmed. Providing more than two levels would be preferable if it’s possible. That’s pretty much it. Labelling some kids as ‘gifted’ and acting like they have some kind of special secret sauce is not necessary or desirable. But if the lessons are too easy, that’s when you get kids who coast and don’t learn to work, or worse check out from boredom and start acting up.
Just out of curiosity, did the kid not know how to set up and solve a linear system?
He teaches elementary school.
But what about the kid who wants to be overwhelmed? Who wants to be extra challenged, and is willing to do the work? Too often, that kid gets left out because we (in the cultural sense) are still really invested in the “secret sauce” model. We need programs that will provide that kid with the support they need to be successful at a higher level than his “aptitude” would suggest. I think the vision is 3 (or 2 or 7) groups of kids all working equally hard, all advancing at their own version of “moderately challenged”. IME, that’s how elementary school thinks about things, because at that age, those kids really do have a lot less personal agency (but not none!). But by high school, you get kids who have a lot more will than aptitude.
Indeed–and I’m not entirely sure how you’d solve this problem with a linear system, or even 100% sure what that means. (It’s been a long time since high school math for me).
Later, I figured you could do something like 25x + 5(28-x)=400, for 20x=260, or x=13 to solve for the number of quarters needed. Is that what you mean? The kids in question hadn’t had any algebra of note since fourth grade, and certainly hadn’t solved anything so tricky, so I wanted to give them a different method.
n + q = 30
5n + 25q = 410
q = (30-n) therefore you can get the expression for a single variable 5n + 25(30-n) = 410 and solve for the number of nickels.
The part that requires the understanding is recognizing how to translate the real-world problem into the algebraic system; everything beyond that is just following steps.
I’m guessing the purpose of exposing the problem to the elementary school kids who aren’t taking algebra is to start them along that kind of thinking.
You’ve basically given two equations, 25x + 5y = 410 and x + y = 30. That’s easy to solve.
ETA: ninja’d by @ZosterSandstorm
Well, sort of. These kids had all done some basic algebra last year, but won’t hit real serious algebra until sixth grade, next year. The problem was a set of unusual problems, part of the Math Olympiad program, that’s designed to get kids to find a way to solve problems of a type they’ve never seen before. It might ask them, “If today is Tuesday, what day of the week will it be in 100 days?” or “If five weights have an average weight of 13 grams, and a 7-gram weight is added to the set, what’s the average of the six weights?” All the problems are solvable with the math they know, but figuring out what math can solve them is the hard part.
They’re great for teaching overall strategies, like “reduce the numbers” and “look for patterns” and “put things in order” and “narrow the possibilities”, but aren’t really designed for teaching specific algorithms: those are taught separately.
So if they’d already learned some, did any of the kids solve the problem with algebra? Or what other methods did they come up with?
Oh, right. I knew that. Sorry!
A better problem is the farm animal problem. Say there are cows, pigs, and chickens. A cow costs $10, a pig $3, and a chicken $0.50. You buy exactly 100 animals. How many of each do you buy?
The thing is there are 3 variables and only 2 equations. For “standard” linear algebra, this is an underdetermined problem with infinitely many possible solutions.
But the important thing to realize is that this is a Diophantine equation, which generally do not have “simple” methods to solve them. Diophantine equations are ones where the answers must be whole numbers, i.e. 2.35 chickens is not part of a valid solution - nobody is going to sell you 0.35 live chickens. There are sometimes tricks you can use but no general solutions. And this is why there is 1 correct solution and not an infinite number of possible solutions.
But you can use a linear system towards a solution. It basically works out to what you did - figure out there are only a limited number of possible solutions and trial and error through them.
I like these kinds of questions, even for kids who haven’t taken algebra yet because they imply there isn’t one “right way” to do math. There’s no formula or whatever, and it is sadly a common misconception that there is always formula or trick that you apply to get the answer rather than thinking through the problem and developing a toolkit of problem solving methods/strategies.
And beyond that, there’s no “right path” to learning math. Agreed with several posters above that we need to get out of the Algebra 1 → geometry → algebra 2 → trigonometry, etc mindset. Linear algebra and basic probability theory/statistics are going to serve them better than differential calculus or trigonometry, for example. For college bound kids who will go into computer science or engineering, those are often more critical while calculus can be saved for later. The ‘classics’ sometimes do need to be updated for modern times. It’s a bit controversial but several of the engineering faculty in my department thought this way. To really get the upper level stuff, calculus is important but we do need to get out of the mindset that there is a well established path towards upper level math. There’s absolutely not.
Or maybe we need to go back to teaching everybody Latin.
sometime in the mid-90s because I remember pre-algebra was offered here in the 7th because my brother took it but he moved back with our dad back east so I don’t know if he took it back there or not in the 8th …
as for me? well I was in the class that was still trying to learn long division in hs …
That was my class. Still, I persevered.
Only 7 states require foreign language classes for their basic diploma at all, and more than 50% of American high schools offer 2 or fewer foreign languages as options.
The “Latin isn’t relevant to the modern world” movement is actually a great demonstration of the dangers of charging into “educational reform” without a clear and justifiable set of goals. Instead of exchanging Latin education for Spanish, Arabic, or Chinese, we exchanged it for nothing. I’m sympathetic to the idea that learning a living language is more valuable than learning Latin, but clearly learning any foreign language has all kinds of benefits. Is “eliminate Latin and replace it with nothing” really progress? How about “eliminate calculus and replace it with nothing?”
No one is saying ‘eliminate calculus’. Any private school can offer calculus in kindergarten if they want to. Every college and university can offer calculus forevermore. At most it is being said don’t use public education tax dollars to shoehorn kids into tracks so that some and only some can start calculus in 10th grade.