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Joined: Sep 2007
Posts: 3,298 Likes: 1
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I just find that these studies that start with "there's a correlation between not understanding fractions and not being able to learn algebra" and go directly to "we need to teach fractions better" may be skipping an important logical step. That's a fair point. The authors make this argument: If students do not understand fractions, they cannot estimate answers even to simple algebraic equations. For example, students who do not understand fractions will not know that in the equation 1/3X = 2/3Y, X must be twice as large as Y, or that for the equation 3/4X = 6, the value of X must be somewhat, but not greatly, larger than 6. Students who do not understand fraction magnitudes also would not be able to reject flawed equations by reasoning that the answers they yield are impossible. Consistent with this analysis, studies have shown that accurate estimation of fraction magnitudes is closely related to correct use of fractions arithmetic procedures (Hecht & Vagi, 2010; Siegler et al., 2011). Thus, we hypothesized that 10-year-olds’ knowledge of fractions would predict their algebra knowledge and overall mathematics achievement at age 16, even after we statistically controlled for other mathematical knowledge, information-processing skills, general intellectual ability, and family income and education. This argument makes sense to me. US elementary schools and math books generally teach kids how to do algorithms and don't go into depth about what's really happening when you do the algorithm, why it works, and how things in mathematics are inter-related. Fractions is a really good example of what I've said. Kids aren't taught about relationships between fractions and division, yet understanding these relationships is critical for doing algebra.
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Joined: Feb 2012
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For all the vilification of Everyday Math that I've seen around here, I have to say that so far, it seems to be doing a good job of keeping fractions tied to division. My daughter has only completed third grade math, though, so there's plenty of time for it to go astray.
And like I said, I don't want anyone to think that I oppose better teaching of fractions. I just think it's valuable so the students will understand fractions better, whether or not it has anything to do with later study of algebra.
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Joined: Feb 2011
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Being taught the material by someone who understands it well beyond the level covered by the class text/curriculum seems pretty crucial to me, too. I think the fact that many primary teachers (and a few secondary ones) don't have a good grip on the underlying math themselves is probably a huge part of this problem. I'm not really surprised by that observation, incidentally. My own mom was a terrible person to be teaching math beyond basic operations. Luckily, she taught K-2, so it was okay. But I had plenty of experience socially with her friends who were no more capable, and some of them were teaching 3-6. 
Schrödinger's cat walks into a bar. And doesn't.
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Being taught the material by someone who understands it well beyond the level covered by the class text/curriculum seems pretty crucial to me, too. I think the fact that many primary teachers (and a few secondary ones) don't have a good grip on the underlying math themselves is probably a huge part of this problem. I'm not really surprised by that observation, incidentally. My own mom was a terrible person to be teaching math beyond basic operations. Luckily, she taught K-2, so it was okay. But I had plenty of experience socially with her friends who were no more capable, and some of them were teaching 3-6. We should stop expecting elementary school teachers to be experts at all subjects and have them specialize by subject, such as English, social studies, science, or math. The current system where students have a single teacher for all subjects makes sense if teachers are regarded as surrogate mothers and not just instructors. Since my eldest has more trouble remembering where his jacket is and putting it on than with fractions, I understand that elementary school teachers play a dual role  .
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the invert-and-multiply algorithm (i.e., a/b ÷ c/d = ad × bc) I do hope it was Val, not the authors, who wrote exactly this...
Email: my username, followed by 2, at google's mail
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I have a copy of the paper Bostonian mentioned today. The working paper version is at http://www.psy.cmu.edu/~siegler/Siegler-etal-inpressPsySci.pdf . From the paper: Marked individual and social class differences in mathematical knowledge are present even in preschool and kindergarten (Case & Okamoto, 1996; Starkey, Klein, & Wakeley, 2004) These differences are stable at least from kindergarten through fifth grade; children who start ahead in mathematics generally stay ahead, and children who start behind generally stay behind. ((Duncan et al., 2007; Stevenson & Newman, 1986). Substantial correlations between early and later knowledge are also present in other academic subjects, but differences among children in mathematics knowledge are even more stable than in reading and other areas (Case, Griffin, & Kelly, 1999; Duncan, et al., 2007). ***************************************************** Fractions are worth teaching for their intrinsic importance, as ElizabethN wrote, but the paragraph above suggests a role for innate ability.
"To see what is in front of one's nose needs a constant struggle." - George Orwell
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And our school has just signed up for Everyday Math, whose curriculum is markedly deficient in . . . arrgh, fractions and long division. I keep telling myself that I can use a tutor and keep my kids up in math, but the elementary school math really does matter. This is just depressing. Good to know though. Thanks Bostonian. There is now a federally-funded Center for Improving Learning of Fractions (!) https://sites.google.com/a/udel.edu/fractions/home with some papers co-authored by Siegler , who also co-authored the paper I mentioned.
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Joined: Sep 2007
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the invert-and-multiply algorithm (i.e., a/b ÷ c/d = ad × bc) I do hope it was Val, not the authors, who wrote exactly this... Ouch. Good catch. I cut and pasted from the paper. I double-checked the final version and a preliminary version written by the authors that I just found. The preliminary version doesn't have the faulty equation (it has no equation). It's possible that this was a mistake made by someone at the journal. I don't know; I wrote to the corresponding author to alert him to the error, and will report back if he replies to me.
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Joined: Feb 2012
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Yes, that was the typo I mentioned (although I see that I was reading quickly and misidentified it as the abstract). I hope there's time to get it fixed.
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Joined: Sep 2007
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Heard back from him already. He's going to ask the journal to fix it. They'll presumably make a new online version of the paper.
Last edited by Val; 08/13/12 01:56 PM.
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