Knowledge of fractions & division predict success in algebra

When I first started writing kitchen table math, with Carolyn Johnston, Carolyn told me that fractions are the math cliff.

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.

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.

More from the article:

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

add this problem to the curriculum

re:

r² is a multiple of 24 and 10. What is the smallest value?

Here, from a few weeks ago, is Stanislaus Dehaene on multiplication:

[O]ur intuition of quantity is of very little use when trying to learn multiplication. Approximate addition can implemented by juxtaposition of magnitudes on the internal number line, but no such algorithm seems to be readily available for multiplication. The organization of our mental number line may therefore make it difficult, if not impossible, for us to acquire a systematic intuition of quantities that enter in a multiplicative relation. (This hypothesis is supported by the fact that patients can have severe deficits of multiplication while leaving number sense relatively intact; in particular, patient NAU (Dehaene & Cohen, 1991), who could still understand approximate quantities despite aphasia and acalculia, was totally unable to approximate multiplication problems).

This passage precisely captures my experience learning arithmetic.

Addition and subtraction make intuitive sense to me; multiplication and division do not. It's really that simple. For me, "math" - math as opposed to simple counting - begins with multiplication and division.

This observation brings me to a corollary: people always say kids fall off the math cliff when it's time to learn fractions, but I think the math cliff comes sooner. I think the math cliff is multiplication, only nobody knows it.

Nobody knows it because falling off a math cliff isn't like falling off a real cliff; with a math cliff, you can walk right over the edge and just hang there for awhile, suspended in mid air, like Wile E. Coyote.

The real drama comes after the fall, which is when kids finally get to fractions. Fractions aren't the cliff, and they aren't the fall. Fractions are the crash-landing at the bottom.

At least, that's my guess for the moment.

Setting metaphor aside, though, if it's true that we are not equipped with an intuitive understanding of multiplication and division, and I believe it is true, why don't more people know this?

Everyone knows fractions are hard; why doesn't everyone know multiplication and division are hard?

It's true most people perceive that certain aspects of multiplication and division are hard. Namely: memorizing the times tables is hard (for many children) and learning to do long division is hard (for many children). But I've never seen anyone take these facts to mean that there is something intrinsically challenging about multiplication and division in a way that is not the case with addition and subtraction.

Why?

I don't know, but I have some thoughts.

Which.... will have to wait. It's getting late, and I'm still trying to edit the body of this post into shape, so I'm going to set that aside and skip to the end, and just say that I think children should probably be taught to solve problems like the one above, which appeared on the October SAT. I'm pretty sure problems this can be used to find out whether students are suffering associative interference between addends and factors, which I bet an awful lot of students are no matter how quickly and accurately they can construct factor trees.

I'm also thinking more attention should be paid to teaching young children the terminology of arithmetic: addends, subtrahends, factors, and the like. I think -- I don't know -- that fluency with the terminology might help reduce associative interference. "All math looks alike": the 5 and the 2 in 5+2 look exactly like the 5 and the 2 in 5x2. But the words addend and factor have nothing in common whatsoever.

More later.

if you can't do it, you don't understand it

cross-posted at the Irvington Parents Forum:

Reviewing his son’s grade-school homework in the early 2000s diverted W. Stephen Wilson from his research in algebraic topology to question basic math education. At two well-regarded private schools, Wilson’s son had encountered the most widely used elementary math curricula, Investigations and Everyday Mathematics. Both encourage the use of calculators for multiplication and division, in line with a 1989 report from the National Council of Teachers of Mathematics that downplayed teaching arithmetic with pencil and paper. “What the schools were doing with math was beyond my imagination,” Wilson says. “I knew that my kid’s third-grade math wasn’t going to prepare him for college.”

A professor of mathematics in the Krieger School, Wilson teaches calculus to undergraduates. In 2006, he decided to conduct an experiment with his Johns Hopkins students. His Calculus I for the Biological and Social Sciences class that year bore close resemblance to the 1989 class. Their scores on the SAT math exam were nearly identical, and the two groups contained the same percentage of freshmen. Curious to see how they’d compare on the same exam, he gave the 2006 students the same 77-point final that he’d given the 1989 class. The results, he believes, confirmed his hunch that students were coming out of K–12 schooling less prepared for college math. When he compared scores by the grading scale in use in 1989, 27 percent of the 1989 students received As on the exam and 37 percent scored Bs. Only 6 percent of his 2006 students would have received As, 26 percent Bs.

[snip]

As another experiment, Wilson gave a short test of basic math skills at the start of his Calculus III class in 2007. The results predicted how students later fared on the final exam. Those who could use pencil and paper to do basic multiplication and long division at the beginning of the semester scored better on the final Calc III material. His most startling finding was that 33 out of 236 advanced students didn’t even know how to begin a long division problem.

Wilson says he wouldn’t be so against calculator use if teachers still taught multiplication and division by hand as well, regardless of the fact that few will ever do math that way as adults. “The theory that people should only learn what they are going to use as adults doesn’t make a lot of sense,” he explains. “If you take that to the extreme, there wouldn’t be much left to K–12 education. If someone is going to fly an airplane when they grow up, should we skip all the intermediate steps and just teach them how to fly an airplane when they are 10?”

[snip]

In 2006 he served as senior adviser for mathematics in the U.S. Department of Education, where he helped form the National Mathematics Advisory Panel. Since his return to his Hopkins classes in 2007, Wilson has continued to review K–12 curricula for various states.

Wilson has found that the brightest students work around what he calls their “unnecessary handicap.” In his study of 2007 Calculus III students, for example, the correlation between being “division clueless” and scoring poorly on the final exam wasn’t as strong as he would have guessed; some of those students did just fine. But that seemed true only for the minority. When he followed up on the class two years later in 2009, one-third of the “division-clueless” students were on academic probation.

Wilson doesn’t like the long-term implications of a new generation of engineers and scientists who can’t divide or don’t know their multiplication tables. He compares it to having car mechanics who only know how to fix automatic transmissions. “You might use a calculator if you’re an engineer, but you need to know what it does. If you need mathematics in your career, then it is probably a good idea to really understand it,” he says. “You don’t understand it if you can’t do it.”

Back to Basics for the “division clueless”
By Lisa Watts
December 6, 2010
Johns Hopkins Magazine

Big "Aha!" moment from the MSMI2010 follow-up day

Saturday 9/25 - late afternoon:

My notes from Wu discussing the teaching of division with remainder:

Actually, this was a big "Duh!" moment.

Prior posts from MSMI2010.

Michael Jackson does long division without a calculator

via Frederick Public Ed

you can't cram math (or anything else)

Numerous parents here have spent years lobbying our high-performing, generously-funded district ($22,000 per pupil spending) to move to the international standard for math education. That being: algebra in the 8th grade.

We have approximately 30% of our 8th graders taking algebra. The figure at KIPP, in the Bronx, is 80%. ($10,000 per pupil spending, roughly)

They're not going to do it. They're so not going to do it they're not even going to say 'no.' They're just not going to do it.

While we're on the subject of well-funded school districts saying 'no,' I should add that the middle school is also not going to allow more students to take Earth Science in the 8th grade. Only forty-eight students, of 150 or so, currently take Earth Science, compared to 100% of students in Pelham. However, in the view of the school that is 48 students too many. As the chair of the science department told us, "If it were up to me, I wouldn't offer accelerated courses to any students in the middle school, but this community demands it."

The district argues, in meetings with parents, that learning depends upon maturity. Not all students are mature enough to learn Earth Science in the 8th grade. Or algebra.

Of course, maturity has nothing to do with ability to learn, as the National Math Advisory Panel reports. However, maturity has everything to do with a student being able to monitor his learning instead of depending on his teacher to perform this function. So, yes. It's easier to teach Earth Science to a high school sophomore than to a student in the 8th grade.

So why do parents continue to lobby school districts across the land to teach serious courses to younger kids?

What is the big deal, after all, about taking algebra in the 8th grade?

What's the difference when you take algebra so long as you get around to it sometime before college?


Brain Rules

It turns out there is a very good answer to that question.

It takes years to consolidate a memory. Not minutes, hours, or days but years. What you learn in first grade is not completely formed until your sophomore year in high school.

Rule # 6: Remember to repeat
John Medina

Bingo.

Here we have one of those facts of life many of us have picked up over the years but can neither verbalize in conversation with school officials nor defend as true, primarily because we don't realize we know it.

We don't know what we know. ^*

On the other hand, when we hear someone else verbalize it, we recognize it as true of our own experience. At least, I did, when I read this statement by James Milgram:

First of all, I claim that taking -- even asking to take it out of the curriculum -- shows a profound ignorance of the subject of mathematics. The point is, in mathematics, many, many skills develop over an extended period of time and are not really fully exploited until perhaps 10, 12, or even 15 years after they've been introduced. Some skills begin to develop in the first or second grade and they do not come to fruition or see their major applications until maybe the second year of college. This happens a lot in mathematics and long division is one of the key examples.

I'm going to guess that this is another reason why Singapore students are so far ahead of American students. Singapore students are doing simple algebra in the 5th grade. They're doing simple algebra in the 5th grade, and they're not dipping in and out of simple algebra, either; they're not being "exposed" to "algebraic thinking."

They're learning what they're learning to mastery.

At age 10.


^* not to be confused with known knowns, known unknowns, and unknown unknowns.

long division in the time of computers

"with apologies to Gabriel Garcia Marquez" (pdf file)

"When we assert that “this is the factorization of a number into primes,” the Fundamental Theorem of Arithmetic is lurking in the background."

One of these days I will be a person who knows what that means.

[pause]

Well, I have now skimmed the entire pdf file and I have no idea what it's talking about, or what the author's views on the place of long division in the curriculum is or is not.

So that was enlightening.


the long version (pdf file)

personal narrative

from Tracy W:

Okay, I love my calculator. Sharp EL-5120. It's on my desk at the moment. It's not much to look at, but its functionality means that it rocks my world. In terms of calculator-adoration I am probably in the top 1% of the world's population. My calculator has literally travelled around the world with me (there's no way I'd trust it to any removal company). I'm not a poet, but if I was I would write love poems to my calculator. The only reason I do not sleep with my calculator is that I fear it will disappear down the end of the bed and I will never see it again. When it comes to using calculators, I strongly suspect I am not normal. However, despite my deep and undying affection for my calculator I am sometimes without it, and on those occasions it is useful to be able to do basic arithmetic such as long division with pencil and paper or in my head. This may not be normal, but why should we educate kids merely to be normal people anyway?

Priceless!

Fractions as Division Problems

My daughter had trouble remember how to find a decimal when given a fraction. She knew it was a division problem, but often transposed the numbers. For example, if she needed to find the decimal equivalent of 5/8 she might divide the 5 into the 8 because it just seems more logical when you are 10 to do it that way. She needed help remembering which way to divide.

Her older (15) brother gave her a mnemonic devise he had learned.

Top dog goes in the house.

There you have it, problem solved. This was new to me too, but it really works, she remembers this easily, even though we only talked about it that one night. Now she never forgets. It's great. Try it with your kids.

The numerator (number on top, or the "top dog") goes inside the "house" the half box you draw for division.

Am I making sense here?

Everyday Math's smoking gun

I want to bring this up front.

LynnG said... Steve, it is always worse than you think.

Here's what EM's Teacher's Reference Manual (Grades 4 - 6) has to say about fraction division.

"Indeed, few adults ever need to divide fractions once they leave school.
Therefore, the main goal of division of fractions in Everyday Math is not to
give students practical skills . . ."

This is an exact quote?

This is an incredibly damning statement to make. I've heard this argument before from regular people, but never so blatently from a math program. I would consider this to be a smoking gun statement. There is no other interpretation.


The simple question is how on earth can they decide on this in K-6! This philosophy guarantees that kids will never need to divide fractions as adults.


Dividing rational expressions like

1/(x-5) divided by y^3/(X+5)

is a required and common skill, even for any high school college math track.


Invert and multiply. What's the big deal? Just think about all of the complicated tasks in life that most kids master. This doesn't rank very high, but oh no! Math is different. It's complicated! It's scary! You have to "understand" it.

It's not about understanding. It's about LOW EXPECTATIONS!

When I talk to other parents about our public schools, they might not know about the problems in math, but they sure know about low expectations.

Everyday Math is based on low expectations. According to Andy Isaacs, it's not for the "elite", and now we know what that means. It means that if you want your child to get into a college math track in high school, you need lots of outside help.