That’s the thing, though…she’s not even at the top of her class. I mean, I personally think she’s a genius, of course. She may indeed have good aptitude for math or she may bottom out by third grade, it’s just impossible to know yet. But this is just how they’re doing it now. Bits and pieces smooshed together - a little algebra, a little geometry, always some arithmetic. I don’t begin to grok the connections or fundamental concepts they’re working on over, say, a week, but it seems to be working.
My son was thrust into the Everyday Mathematics with second grade (we moved into the district them) through eighth. He was reading over my shoulder when I wrote the earlier post. He says it wasn’t hard, it was just different than how he remembers his first couple of years in the more traditional curriculum and how his high school math (again, traditional, first algebra, then geometry, etc.) He’s now doing just fine in…um…whatever comes after calculus in college. (He’s 19.)
Don’t get me wrong: some people HATE this way of teaching math. Some of them are mathy people who know mathy math stuff (which I don’t). It’s been years of controversy bringing it into our schools. But I’ve been nothing but pleased, although we do add a little drill ourselves. Keeps us entertained on long car rides.
runs screaming from thread
(No, seriously, it’s just ad+ae+af+ag+bd+be+bf+bg+cd+ce+cf+cg, right? I think? Everything multiplied by everything and added together?)
(Bold added).
Ahh, if only I had known that back in my school days. Especially in Calculus (which, to be sure, isn’t 10th grade stuff). All those pages and pages of integrals I busted my ass memorizing (ass being where my brain must have been). And now you tell me, I didn’t really need to memorize anything. All I needed to do was provide the right answer.
If only I had known then that it was all so simple. Leaffan, where were you when I needed you?
Uhhh… This is a whoosh, right?
P.S. You didn’t actually say what your approach is to multiplying polynomials. But if you do (a+b+c)(d+e+f+g) the same basic way you do (a+b)(c+d), then I’m not quite sure if you’re actually using the FOIL rule. I sure tutored a lot of students who tried FOIL on problems like that and were lost.
We weren’t talking calculus. We were talking simple percentages. If you think you need to memorize formulae to decide how much more, or how much less, or what the increased percentage is then you’re not able to interpret the question correctly.
Interpreting the question is what math is all about really. For the gasoline question I deduced that the price went up by 5 cents. The original price was $2.99. I divided 5 cents by 3 in my head to get the correct answer as 1.67.
I don’t think most people can grasp simple math in this manner. Can you? And I’m serious and not being snarky.
The FOIL rule just sounds needlessly complicated to me (I had to Wiki it, hadnt heard of it). But I probably would have got Craftermans % problem wrong, since I would have divided something by something but they may or may not have been the right somethings.
I wouldn’t go that far. But I might say that some, perhaps many, primary school teachers are bad at math and they can’t effectively teach what they don’t really understand. So “teaching the concept” fails in the classroom because the teacher doesn’t have a strong grasp on the concept themselves. If students get a few years in a row of muddled teaching, it could be very difficult for them to get back on track.
question –> the price of gas today is _____% higher than yesterday
replace with values –> 3.04 is 0.05 higher than 2.99
convert answer to a percentage –> 0.05 as a percent of 2.99
why 2.99 instead of 3.04? because we’re comparing the 0.05 to yesterday and not today (than yesterday)
The problem is that they don’t understand *why *they’re “FOILing”. So they follow “the rule” very literally, even when it doesn’t apply. “F means First…okay, that’s ad, got it. O means Outside…okay, ag. I means…uh…crap!..what’s the Inside?!..THERE’S MORE THAN ONE INSIDE!..OMG!..I’M SO STUPID, MATH IS HAAAAAAAARD!!!”
It’s not hard, they’ve just been sabotaged by being provided with a “shortcut” before they’ve understood the lie of the land. When the shortcut doesn’t precisely fit, they don’t know how to adapt the directions and get to the right destination.
Among a group of students taught FOIL without grokking the concept, you’re most likely to get: (a+b+c)(d+e+f+g) = ad+ag+cd+cg
Whereas (assuming I got it right) I haven’t remembered FOIL in 20 years, nor have I used factoring (this is factoring, right?), but I somewhere along the way formed the sloppily worded/remembered concept “Everything multiplied by everything and added together,” so I could still do it with more than binomials. If I had learned a conceptual level deeper and understood why the [BLEEP] we wanted to factor and what it really meant, I’d probably not even have the haze of self doubt about my answer that I do. But I never quite really got to that point. I feel like I memorized a list of vocabulary and a few grammar rules, but never got to the point of being fluent in math as a language. I regret that.
My kids got more math than I ever did - I graduated in 1984 and the highest math course my high school even offered was trig. My daughter is in seventh grade and is doing the math I did in tenth grade. She is on track to take calculus as a Junior. If she takes math her senior year, she’ll need to go to the University of Minnesota for a math class. There are about 20 kids out of 200 on this path.
My son is in advanced math as an eighth grader doing the math I did in ninth grade. He’s tracked to take calculus his Senior year.
They’ve been doing “spiral math” so very little emphasis on memorization, but a lot of emphasis on conceptual math - understanding why you subtract x from both sides, not just doing it.
However, my kids elementary school was failing no child left behind, particularly in math. We had lots of resources going towards trying to get everyone competent and few going towards kids who are adept. And that is what I remember from my own school - there were a number of kids who even in Senior High were relearning “long division” - their capacity for math had been reached.
“FOIL” is, properly speaking, not a rule but a mnemonic device, handy for helping you remember how to apply the general rule in one particular common kind of situation. That doesn’t make it bad, any more than “Every Good Boy Does Fine” is bad because it only applies to the treble clef.
As to memorization: there’s relatively little in math that has to be memorized, in the sense that there’s relatively little that’s arbitrary. Things like multiplication facts can be figured out. Anyone who understands the underlying concepts could figure out what 8 x 4 is, and why, if they didn’t already have the correct number memorized. (And if they somehow had the incorrect number memorized, you could explain to them convincingly why it was wrong.) But it’s an enormous handicap to have to stop and figure things like 8 x 4 out (or to reach for a calculator) every time you need them, or (in a different context) to have figure out what numbers multiply together to give 32, so having them committed to memory is in fact important.
I just took an algebra class at the college level. And I was really impressed and happy by how much more they used real world applications, instead of just grinding out problem after problem.
I’d say they are teaching maths better nowadays then when we were kids. I had a really hard time with math when I was a kid/teen. I got a 490 on my maths section on my SATs.
I just brought down an A- in my algebra course. Now algebra is always one of the courses that made sense to me, but this was home study, and the textbook had a myriad ways to explain it and understand WHY things were important.
And you can pry FOIL from my cold dead hands, as well as PEMDAS. Some of these little memory devices are AMAZING.
He’s got some really insightful things to say, but I have no confidence that the way he teaches can be implemented as a teaching method. It relies too much on creative expertise. The very thing that makes it awesome also makes it impossible to put reliably into practice.
He also does say some things that strike me as odd. He’s bothered that students are asked to prove that opposite angles at the intersection of two straight lines are equal. His problem is that this is so blindingly obvious, and can serve only to make students mistrust their own intutions. But I find that one of the strengths of education in a liberal society is that it helps students see that what seems intuitively obvious to them can nevertheless be supported. Moreover, it helps us learn that things that seem intuitively obvious may not be true. I’m not sure what’s supposed to be wrong with getting students to mistrust their own intuitions. Learning to mistrust my own intuitions has been a source of considerable growth and satisfaction.
And then he gives the example of the proof that a triangle with one side being the diameter of a square is always a right triangle. The student proof he likes starts with the statement (words to this effect) “Rotate the triangle around so it’s exactly opposite inside the circle.” The guy likes this proof, and I can see that it has the right idea, but that statement is so vague! I have no idea why we shouldn’t ask the student to clarify it and prove that it can be done.
See, I was never taught FOIL or PEMDAS, which is why I have a different approach to solving the problems. I had to look up FOIL, I still haven’t looked up PEMDAS.
ETA: Order of operations huh. We were just taught the order of operations. Never had an issue in my life with it.
The teaching of math at a young age was never something I had to really think about. I tutored my peers in high school and while those that were not that good at math were missing basic skills, they still had some knowledge (of at least arithmetic) so we focussed on the repetition of new skills.
That all changed when my daughter came to live with us. She is bad at math. Well, actually, she was so far behind that there was no chance of her ever catching up. She has no learning disability and is certainly smart enough to understand things.
(Small background, I am a Mathie. I have a Bachelor of Mathematics. My husband is Mr. Statistics and Arithmetic. Neither of us ever struggled with math.)
We have made it our overreaching goal to catch our kid up.
Here are some things we have learned:
As long as your child is passing (as in earning 50%), there will be no extra help or intervention. (Even when it is obvious that she doesn’t even understand what subtraction means in grade 4.)
If your kid is diagnosed with any sort of disability they will get extra help regardless of how well they are doing.
Primary teachers who fundamentally understand math are few and far between.
There are two strategies to teaching math: understanding and rote. The schools try to use understanding and throw out the repetition. However, that repetition is needed for a child who is at all struggling to learn mathematical concepts.
Calculators are used to replace learning of facts like times tables.
It became our job to fill in the holes that we saw in her learning. Here is what we do:
Every night we review her math schoolwork from that day. We ensure that she can both do the work and understands what she is doing. This takes about an hour a day.
She does Kumon 6 times a week. Their entire strategy is based on rote learning so she is getting faster and better at basic addition, subtraction, multiplication, etc. This makes doing the schoolwork above much easier.
We have a tutor that comes in once a week to fill in the holes in understanding of basic numbers that she missed in the years she was neglected. He is mostly there to get math to be fun. He rocks.
She is not allowed to use a calculator at home.
We work simple math into daily activities (time, cooking, etc.)
So, with all that in mind, what I really see in the schools is that we need to:
Devote more time to math in the early years.
Focus on both understanding and rote learning in mathematics. The understanding is so concepts can be built on and the rote is so that when things get arithmetically more difficult, they are not onerously time consuming.
Give kids who are struggling help regardless of no diagnosis.
My biggest problem with all this is that we spend (quick math) 10 hours a week on math at home. They spend (more math) 4 hours at school. There is no way that we would be making the gains we are seeing with only 4 hours of math instruction.
(On the bright side, she was at about an early grade 2 level in math when she started this year (grade 4) and now she is about a late grade three. There is reason to believe that by the end of the year, she will be at grade level.)
And none of them have done anything to improve our students’ scores compared
with students from the rest of the world.
Next time you post a 99-page pdf citation please post some extracts so your readers
don’t have to slog through the whole thing themselves to get some idea what it is trying
to convey.
I scrolled to almost the end looking for sample problems before finally finding a set
of four on page 90. There was a ridiculous typo (can you find it?- it’s obvious), and
I am discouraged that the authors could not even get their boilerplate exactly right.
Also, I doubt they really needed 90+ pages to get their message across.
I had the advantage of rigorous private schooling K4-12, and if the samples on p90
are typical, then I think I probably was being taught at that level in Grade 8 (not that I found it easy!),
although I cannot recall specifics with certainty. FWIW I was a mediocre mathematician.
I scored 527 SATM in the fall of 1965 and 597 SATM in the fall of 1966 (but: NO CALCULATORS!).
In college I earned a B- in Algebra I and a C- in Probability I and then left academic mathematics for ever.
What does “Everyday Mathematics” mean? The problem set I mention above involved
graphs unlikely to be encountered in the course of a normal day except by some of those
working in technical professions.
I expect your daughter has much above average mathematical aptitude and could learn
the subject no matter how ineptly it is taught. Maybe she will be lucky and the Chicago
school system will turn out to be one of the better ones in the country. If so, however,
proof will have to come from the system-wide averages rather than the progress of a gifted student
such as your daughter.
She may indeed have good aptitude for math or she may bottom out by third grade, it’s just impossible to know yet. But this is just how they’re doing it now. Bits and pieces smooshed together - a little algebra, a little geometry, always some arithmetic. I don’t begin to grok the connections or fundamental concepts they’re working on over, say, a week, but it seems to be working.