Today in their final report, the
National Math Advisory Panel said:
Difficulty with the learning of fractions is pervasive and is an obstacle to further progress in mathematics and other domains dependent on mathematics, including algebra. It also has been linked to difficulties in adulthood, such as failure to understand medication regimens. Algebra I teachers who were surveyed for the Panel as part of a large, nationally representative sample rated students as having very poor preparation in “rational numbers and operations involving fractions and decimals” (see Panel-commissioned National Survey of Algebra Teachers, National Mathematics Advisory Panel, 2008). The Final Report of the National Mathematics Advisory Panel, p. 30
Today in the blogosphere, a
student teacher said:
Today I taught a lesson about fractions from Everyday Mathematics. Fractions were not something I was good at in school, so I was nervous about teaching it. It went okay, but some of the students gave answers that were not correct, and I was having trouble explaining why they were correct. Luckily my master teacher was in the classroom so she was able to help me give an explanation on why that students answer was not correct.
Houston, we have a problem.
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14 comments:
As one of my college professors once told me:
When a student is having trouble with calculus, they're really having trouble with algebra. And when a student is having trouble with algebra, they're really having trouble with arithmetic, usually fractions.
wow!
I love that!
I don't think you were around back when Carolyn and I started the first ktm. I knew nothing about math & math ed; I didn't even know math was cumulative.
And btw this is one of the ways schools get away with early tracking. My district put C. into "Phase 3" math when he was in 3rd grade. I had no idea that a Phase 3 placement meant he couldn't take calculus in high school, and nobody told me.
The district made a binding decision about his future right then and there without consulting the family or explaining to us what that decision meant.
It got worse after that.
When the school brought in Trailblazers they actually couldn't tell parents whether their kids would be prepared for calculus in high schol.
They had no answer at all!
Back on topic: when I first met Carolyn she told me that the HUGE PROBLEM in K-12 math was fractions.
I was amazed; I had no idea was talking about. I'd learned fractions well enough when I was a kid; I'd learned long division easily and had no clue long division was harder than addition, subtraction, or multiplication. All these things seemed on a par.
After that I went around asking math professors about their students and it got to be a joke because across the board they would say, "My students can't do fractions."
Of course, the funny thing was that the ONE traumatic moment I remember from K-6 was a fraction assignment in 2nd grade where we had to color in the number of parts represented by a fraction and I did it wrong. I was horribly upset.
And the reason I became C's math reteacher was that - you guess it - he had failed all of the fraction lessons & tests in 4th grade!
I'm still vigilant about fractions. I'd say he has a decent grasp of the concept now but I'm always thinking about it - about ways to make it clearer to both of us - and I'm always looking to see whether he can do word problems involving fractions.
His teacher also gives problems involving fractions. I think he tells them, "no fraction prejudice," something like that. That made a big impression on C.
"...when I first met Carolyn she told me that the HUGE PROBLEM in K-12 math was fractions."
Thank you again for all of your work. KTM has helped lots of parents and kids, and, hopefully, affected the national debate in some small way.
I was very happy to see the Panel's emphasis on fractions, especially in light of a forthcoming book by Dennis DeTurck, an award winning math professor at Penn and new dean of its College of Arts and Science.
Have you come across any of DeTurck's proclamations? He believes, for example, that grade schools should stop teaching fractions and make do with decimals and calculators.
With his credentials, Dean/Professor DeTurck has the power to inflict tremendous harm on math education. Indeed, he recently received a huge NSF grant for teaching training-- yikes!
I recently blogged on my take on fractions, including an exchange I had with DeTurck.
(http://oilf.blogspot.com/2008/03/why-teach-fractions.html)
an award winning math professor at Penn and new dean of its College of Arts and Science
swell
I suspect he himself is perfectly capable of adding, subtracting, multiplying, and dividing fractions
This is directly analogous to native English speakers deciding that Hispanic kids should take all their classes in Spanish
It's not what the parents want, but hey --
Thank you again for all of your work. KTM has helped lots of parents and kids, and, hopefully, affected the national debate in some small way.
right back at you!
The one thing ktm has definitely done is to provide aid and comfort, not to mention knowledge and resources, to parents and others trying to opt out of constructivist math.
C. is in the shape he's in - which, these days, is pretty good - largely due to ktm. He has a good class this year but he wouldn't have made it to that class without ktm.
Gosh!
That reminds me.
When I first started writing ktm my goal was to get C. to algebra in 8th grade.
Now he's not only there he's doing well.
I plan to stay vigilant on the math front until he's out of h.s. I imagine he's going to have terrific teachers next year but math really is different from every other subject.
I've come to think that schools should be doing fine-grained formative assessment of math knowledge for each and every student they've got.
I probably feel the same way about reading though I haven't thought about it...
Yep, fractions are tough for kids, and the 6th grade textbooks I've looked at aren't really good enough at teaching fractions. Most beginning teachers haven't had the opportunity to really think about fraction learning in detail either. It's a problem. And no, I don't think that removing fractions (and all other "hard" things) from the curriculum is a very good solution to the problem.
I'm still not very good with fractions. If there's another way to tackle a problem I'll use it.
Now that's constructivist!
Thinking about fractions in detail WON'T HELP if you don't understand mathematics.
That's the problem. The more the kids think about them, the more confusing it seems. The more the teachers think about them, the more they get confused. Why? Because they have no grounding in how fractions are NUMBERS, and the rules of arithmetic apply coherently to them the way they do to whole numbers. so they confuse themselves with new analogies, with new abstractions, instead of seeing that the fractions behave just as reasonably as numbers they do know.
There's a reason that the National Mathematics Advisory Panel is trying so hard to address this; it's the first systematic failure in American mathematics education, and it's the foundational failure upon which no other structure can be built.
I'll go back and find more of Prof. Wu's papers explaining fractions with clarity in the next day or so...
There are many reason that fractions should not be junked in elementary school. One basic reason is that old favorites like 1/3, 1/6, 1/7, 1/9, and, frankly, most fractions, do not have a finite decimal representation. If you just use decimals and calculators you are left with the falsehood that 1/3 = 0.3333 (or as many 3s as your calculator displays). You can't really understand decimal expansions without understanding limits. If we are going to put fractions off until calculus, then we might as well go ahead and put decimals off until calculus,too. (I'm not sure what you can do until then.) And then when you do get to calculus, and start with secant lines (f(a) - f(b))/(a-b) I guess the first lecture with be "consider the ratio of 1 and 2 ....", by way of introduction.
I'll go back and find more of Prof. Wu's papers explaining fractions with clarity in the next day or so...
I've found Wu's materials pretty challenging to read, so if you can translate them into slightly more accessible language -- or just give us a gloss -- that would be a big help.
I do need to finally go back to the fraction paper.
I think I'll see what Ron Aharoni has to say about fractions first.
Wrongheadedness can cut many ways. I know a local high school teacher who considers himself very anti-constructivist but thinks that fractions shouldn't be taught until high school.
I forgot to add - I haven't seen him teach so I can't address his methods, but he says he's anti-constructivist and uses traditional, non-reform influenced methods.
"I've come to think that schools should be doing fine-grained formative assessment of math knowledge for each and every student they've got.
I probably feel the same way about reading though I haven't thought about it..."
We had a neighbor in Alabama who taught during their Reading First implementation. She was a big fan of phonics already, she was just switched to a new Reading First subsided phonics program. It had frequent testing (weekly testing?) of each student as part of the program. She taught in an inner-city, majority-minority school. She said it was a lot of work testing the students that often, but that the frequent testing caught a lot of students who would have slipped through the cracks in the past and enabled them to be identified and brought up to speed.
We've lost track of them, we've moved 2 times since then and they've moved at least once, or I'd call and find out what program they were using and how often they tested. I can't remember, she told me at the time, but it was a program I had not heard of before so it did not stick in my brain.
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