Yesterday, Glen left a link to a new study in Psychological Science confirming the critical importance of fractions -- and long division -- to a child's future success in algebra:

Our main hypothesis was that knowledge of fractions at age 10 would predict algebra knowledge and overall mathematics achievement in high school, above and beyond the effects of general intellectual ability, other mathematical knowledge, and family background. The data supported this hypothesis.More from the article:

and:

Early knowledge of whole-number division also was consistently related to later mathematics proficiency.

and:

The greater predictive power of knowledge of fractions and knowledge of division was not due to their generally predicting intellectual outcomes more accurately.

ABSTRACT

Identifying the types of mathematics content knowledge that are most predictive of students’ long-term learning is essential for improving both theories of mathematical development and mathematics education. To identify these types of knowledge, we examined long-term predictors of high school students’ knowledge of algebra and overall mathematics achievement. Analyses of large, nationally representative, longitudinal data sets from the United States and the United Kingdom revealed that elementary school students’ knowledge of fractions and of division uniquely predicts those students’ knowledge of algebra and overall mathematics achievement in high school, 5 or 6 years later, even after statistically controlling for other types of mathematical knowledge, general intellectual ability, working memory, and family income and education. Implications of these findings for understanding and improving mathematics learning are discussed.

[snip]

Marked individual and social-class differences in mathemat- ical 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). There are substantial correlations between early and later knowledge in other academic subjects as well, but differences in children’s mathematics knowledge are even more stable than differences in their reading and other capabilities (Case, Griffin, & Kelly, 1999; Duncan et al., 2007).

These findings suggest a new type of research that can con- tribute both to theoretical understanding of mathematical development and to improving mathematics education. If researchers can identify specific areas of mathematics that consistently predict later mathematics proficiency, after controlling for other types of mathematical knowledge, general intellectual ability, and family background variables, they can then determine why those types of knowledge are uniquely predictive, and society can increase efforts to improve instruction and learning in those areas. The educational payoff is likely to be strongest for areas that are strongly predictive of later achievement and in which many children’s understanding is poor.

In the present study, we examined sources of continuity in mathematical knowledge from fifth grade through high school. We were particularly interested in testing the hypothesis that early knowledge of fractions is uniquely predictive of later knowledge of algebra and overall mathematics achievement.

One source of this hypothesis was Siegler, Thompson, and Schneider’s (2011) integrated theory of numerical development. This theory proposes that numerical development is a process of progressively broadening the class of numbers that are understood to possess magnitudes and of learning the functions that connect those numbers to their magnitudes. In other words, numerical development involves coming to understand that all real numbers have magnitudes that can be assigned specific locations on number lines. This idea resembles Case and Okamoto’s (1996) proposal that during mathematics learning, the central conceptual structure for whole numbers, a mental number line, is eventually extended to rational numbers. The integrated theory of numerical development also proposes that a complementary, and equally crucial, part of numerical development is learning that many properties of whole numbers (e.g., having unique successors, being countable, including a finite number of entities within any given interval, never decreasing with addition and multiplication) are not true of numbers in general.

One implication of this theory is that acquisition of fractions knowledge is crucial to numerical development. For most children, fractions provide the first opportunity to learn that several salient and invariant properties of whole numbers are not true of all numbers (e.g., that multiplication does not necessarily pro- duce answers greater than the multiplicands). This understanding does not come easily; although children receive repeated instruction on fractions starting in third or fourth grade (National Council of Teachers of Mathematics, 2006), even high school and community-college students often confuse properties of fractions and whole numbers (Schneider & Siegler, 2010; Vosniadou, Vamvakoussi, & Skopeliti, 2008).

This view of fractions as occupying a central position within mathematical development differs substantially from other theories in the area, which focus on whole numbers and relegate fractions to secondary status. To the extent that such theories address development of understanding of fractions at all, it is usually to document ways in which learning about them is hindered by whole-number knowledge (e.g., Gelman & Williams, 1998; Wynn, 1995). Nothing in these theories suggests that early knowledge of fractions would uniquely predict later mathematics proficiency.

Consider some reasons, however, why elementary school students’ knowledge of fractions might be crucial for later mathematics—for example, algebra. If students do not under- stand fractions, they cannot estimate answers even to simple algebraic equations. For example, students who do not under- stand 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.

Early Predictors of High School Mathematics AchievementRobert S. Siegler1, Greg J. Duncan2, Pamela E. Davis-Kean3,4, Kathryn Duckworth5, Amy Claessens6, Mimi Engel7, Maria Ines Susperreguy3,4, and Meichu Chen4Psychological Science 23(7) 691–697

## 16 comments:

there's a reason why ability with fractions predicts success in algebra.

Algebra is abstract. It is about understanding how definitions and rules lead to results. Fundamentally. you can´t count your way to finding all of the solutions to y= ax^2 + bx+C.

Arithmetic is not abstract per se. Pedagogically, you can introduce abstraction to arithmetic to help students begin to think mathematically, but they can fallback on manipulatives, counting, etc.

Fractions are the first abstractions. You can't get farther than 4th s or 6th s with fraction manipulatives, and you can't handle general fractions except abstractly. To work fractions, you need to work definitions, symbols, procedures and you work it by coherent reasoning. If you can't do that for fractions, you won't be able to do it for algebra.

This is why it's to important to teach the reasoning and meaning behind the procedures in fractions, to connect the dots for kids to see how math behaves.Students must be given a chance to work symbolically, to recognize definitions, to reason. Fractions must be taught with substance, depth, and difficulty.

To toot my own horn, you can learn more about fractions here:

www.msmi-mn.org

Our 10 year old is struggling with exactly that transition. Sometimes, he seems to get stuck in the abstraction and I can't get him back into the real world, and other times he rejects it completely and gets very frustrated trying to force the abstraction into reality. Yesterday he had to tell me how many kilograms were in 1/5th of 2kg--a problem like ones he had done dozens of times before. He kept giving me the abstraction: essentially 20%, when I wanted a real physical, measurable answer: 400grams. The day before, he had that reversed and couldn't figure out how to give me the abstract answer.

He's almost made it through, though. Singapore's handling of fraction division was very good, and I always emphasized the idea that you were trying to find out how many of the divisor could fit into the dividend.

I think he's going to struggle a little with algebraic thinking, when letters with infinite possibilities replace fixed numbers. The cause of his problems with fractions will be the exact same as any problems he has with algebra: moving into the abstract world.

Allison: "There's a reason why ability with fractions predicts success in algebra.

Algebra is abstract.

...

Arithmetic is not abstract per se.

...

Fractions are the first abstractions. You can't get farther than 4th s or 6th s with fraction manipulatives, and you can't handle general fractions except abstractly."

Thank You!

I've been working off the assumption that the key to algebra was fractions, but didn't have a good explanation *why* this was the case. You have just provided an explanation that makes sense.

It also (sorta) explains the long division correlation ... with large enough numbers, you can't fall back to counting or beads.

-Mark Roulo

Ann,

One thing that really helps is putting fractions on the number line, and explicitly defining them in terms of the number line.

A fraction K/L is the point on the number line when you break the unit length into L equal length parts and take K. You need a little more exactness in the definition, but not much. See these posts for more.

http://kitchentablemath.blogspot.com/2008/03/on-our-way-to-fractions-number-line.html

http://kitchentablemath.blogspot.com/2008/03/more-fun-with-number-lines-and.html

the best source though is Wu's book:

http://www.amazon.com/Understanding-Numbers-Elementary-School-Mathematics/dp/0821852604/ref=sr_1_1?ie=UTF8&qid=1344740656&sr=8-1&keywords=wu+elementary

the number line is an abstraction that he can use over and over and over again.

Ask him to use that definition to find 1/3, 4/3, 0/3, 100/5.

then have him practice with decimals: this is something Primary Math does poorly, in fact. Decimals are fractions whose denominators are a power of ten, so have him locate 1/10, 1/100, 10/100, 103/100,

then, after you have hin get used to the number line, you can connect it to concrete problems using bar models.

the number line is how to express a fraction, then match that up against the bar to find a given fraction of a set.

Mark,

You're welcome!

And because fractions are the first abstractions, when textbooks and schools deliberately shy away from that abstractness, and resort to analogies or pizza slices, they are hampering the ability of their students to reach algebra. The shallow curriculum in fractions means the steep slope to algebra that many students can't negotiate.

My guess is division with remainder for even slightly large problem size requires putting together all of the pieces of arithmetic quickly. Mastering math facts is a prerequisite to doing division with remainder quickly, yet math facts can be mastered without understanding, and the overall maturity needed to do the algorithm requires understanding beyond the facts.

Hee hee. The cheapest price for Wu's book on Amazon is $77!!

He's good with decimals, it was just jumping between abstract and tangible that he was getting stumped on. The arithmetic part he understood.

Music is an excellent lead-in to fractions. When my boy got to them it was old hat already (of course a quarter is half of a half).

Ann,

It's *because* he´s good with decimals and the procedural arithmetic that you should use it as a starting point for bridging the abstraction in fractions.

So have him not '"ump" between, but use his knowledge to facilitate understanding fractions from the definition, with the number line as the bridge.then you can explain an area model for fraction multiplication and division.

Kids need to understand why the rules they know are true: why can you drop zeroes at the right end after a decimal point, but not the left, why do we invert and multiply, why we implicitly rely on constant rate, etc. Manipulating the rules is not enough.

Allison,

Given that fractions are more abstract than the preceding arithmetic taught, isn't it true that no matter how well fractions are taught, some students will never grasp them and will not grasp algebra, either, because they cannot think abstractly?

"These findings suggest a new type of research that can con- tribute both to theoretical understanding of mathematical development and to improving mathematics education."

Yes, when common sense is completely lacking.

Of course, we all know exactly what:

"knowledge of fractions at age 10"

means.

Bostonian, I agree with a lot of your emphasis on IQ differences as a major contributing factor to many problems. Even so, almost all human beings, even those with low IQs, can think abstractly. I'm not saying that all human beings are EQUALLY capable of abstract thinking, just that I doubt that a categorical inability to think abstractly exists. If it does, it would be a rare (<1%) condition.

The architecture of the brain causes us to blend specific examples with common features into generalizations about anything with those features. That's abstraction. People who's brains don't do that would be quite exceptional. It's harder for some people to learn to recognize certain features, and it may take many more examples before they do, but everyone can do this to some extent.

Of course we all do it to DIFFERENT extents, and a weakness at one level of abstraction will be a weakness squared at the next. Those who form abstractions more slowly may be moved on too quickly to the next level, where they'll fail utterly. But it's not because they have "no ability to think abstractly."

And, of course, there are lots of other reasons people learn too slowly, get moved on too quickly, and end up with a weakness squared in algebra, and drop out. Ineffective pedagogy, for example....

This is why we've been in Kumon since 5th grade. School did a fine job teaching arithmetic but the slippage started with fractions!

Catherine,

I know that this comment has nothing to do with the content of the post, but I just wanted to point out that there's a subject-verb agreement error in the title: knowledge (subject, singular) - of fractions and division (prepositional phrase) - predict (verb, plural). It should be "predicts."

Erica

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