Three sample problems from the PRAXIS 1 Content Assessment college students entering the field of education are frequently required to take:

1. Which of the following is equal to a quarter of a million?
a) 40,000 b) 250,000 c) 2,500,000 d) 1/4,000,000 e) 4/1,000,000


2. Which of the following fractions is least?
a) 11/10 b) 99/100 c) 25/24 d) 3/2 e) 501/500


3. Which of the sales commissions shown below is greatest?
a) 1% of $1,000 b) 10% of $200 c) 12.5% of $100 d) 15% of $100 e) 25% of $40

The Educational Testing Service (ETS) describes these problems thus:

The Pre-Professional Skills Test in Mathematics measures those mathematical skills and concepts that an educated adult might need. It focuses on the key concepts of mathematics and on the ability to solve problems and to reason in a quantitative context. Many of the problems require the integration of multiple skills to achieve a solution. [snip] Computation is held to a minimum, and few technical words are used. Terms such as area, perimeter, ratio, integer, factor, and prime number are used, because it is assumed that these are commonly encountered in the mathematics all examinees have studied. Figures are drawn as accurately as possible and lie in a plane unless otherwise noted.

see also: MathInSalinaKansasPart2

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Here is Saxon's explanation of the curriculum:

Saxon Math . . . systematically distributes instruction and practice and assessment throughout the academic year as opposed to concentrating, or massing, the instruction, practice and assessment of related concepts into a short period of time -- usually within a unit or chapter.

I can vouch for this.

SAXON 6/5 has 120 lessons in all, plus 12 'Investigations' & 3 Appendix lessons, and when you get to Lesson 120 you're still practicing the stuff you learned back in Lesson 1.

There are 100 or more problems and computations in each of the 120 lessons: Fast Facts, Mental Math, Problem Solving, Lesson Practice, and, finally, Mixed Practice.

This is what we call drill and kill.

Cognitive psychologists call it automaticity:

Practice Makes Perfect But Only If You Overlearn Ask the Cognitive Scientist: How We Learn by Daniel T. Willingham

review

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Daniel T. Willingham's 'Ask the Cognitive Scientist' columns for AMERICAN EDUCATOR (wonderful)

William Schmidt, et al's phenomenally helpful 'A Coherent Curriculum: The Case of Mathematics' (Schmidt headed the Third International Mathematics and Science Study (TIMSS), and summarizes his findings here.)

Specific Learning Disabilities: Finding Common Ground. A Report Developed by the Ten Organizations Participating in the Learning Disabilities Roundtable. This is the American Institutes of Research 2002 consensus report: what findings, hypotheses, and theories do 10 different organizations and insitutions, including the Department of Education and the Learning Disabilities Association of American, agree to be true of 'specific learning disabilities.' (I haven't read this yet.)

See also: PracticeAndOverlearningPart1

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I am familiar (very familiar) with Everyday Math, and it has clear weaknesses that we'll discuss at length in time, but I was struck by the following quote in today's article:

Klein said that as a result of whole math programs such as EM, CSUN and other colleges must offer entering freshmen remedial math classes at a level as low as third grade. He said he’s seen, for instance, calculus students who can’t add fractions.

"This is kind of the lost generation, ruined by these liberal-minded policies," Klein said. "The truth of the matter is it’s just a crummy program."

It may be a crummy program -- I have certainly found it hugely frustrating to work with -- but it wouldn't be fair to blame Everyday Math for the existence of vast numbers of calculus students who can't add fractions. The problem has been around a lot longer than Everyday Math has.

I taught at SUNY Binghamton in the early 80s, and we had plenty of calc students who couldn't add fractions. When I was a grad student at Louisiana State University, the remedial math caseload on the mathematics department was so heavy that a whole class of 'instructors' -- essentially the equivalent of high school teachers in schooling and training -- were employed by the math department to teach remedial math classes, and a typical grad student was assigned full responsibility for 2 classes of remedial math every semester. That's more than 60 students per grad student.

And these classes were serving just the students who had been identified as needing remedial math classes; many slipped through the cracks. You bet a lot of the students in LSU's calculus classes couldn't add fractions. Nor is the problem confined to LSU; public universities everywhere, with few exceptions, have large remedial math loads. It's been going on for at least twenty years, long before Everyday Math appeared on the scene.

I don't think there are any simple explanations. But I do think we're floundering, and we need to look to countries with a better track record for guidance.

Furthermore, any math professor can point to plenty of failures in math education within his own experience, but individual failures don't help to explain what we're doing wrong at the policy level. For that, we'll need sound research.

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Answer: really far.

In 1923, the NEW YORK TIMES reported that fewer than half of seventh grade students could convert the fraction 1/5 into a decimal.

The Columbia Teachers College had a plan.

The new aim of the progressive arithmetricians is to abandon drilling in artificial problems and to bring mathematics close to every-day life.

from: 'New Teaching Puts Life into Dreary Arithmetic', NYTIMES December 9, 1923

Apparently, the plan was working.

The new method is so successful, according to its sponsors, that one school has playfully threatened to abandon it for the reason that the pupils are so enthusiastic over arithmetic that their teachers can scarcely interest them in other subjects.

This was the start of progressive education in America.

So flash forward to 1989, and we find NAEP reporting that 60 percent of seventh grade students can 'express simple fractions' as decimals.

A mere 70 years of progress, and 10% of American seventh graders who wouldn't have known that 1/5 is the same thing as 20% back in 1923 do know in 1989.

That was my first thought.

My second thought was, OK, I'll take it. 10% is 10%.

Then I noticed Chris Correa's second post on the subject.

I browsed through the publicly released NAEP questions and found the most comparable question to be from 1992: Of the following, which is closest in value to 0.52?

A) 1/50
B) 1/5
C) 1/4
D) 1/3
E) 1/2

Only 51% of eighth-graders correctly answered this question. Nearly 30% of students responded that 1/50 was closest in value to 0.52.


This is my beef with constructivism.

It's not like constructivism hasn't been given a fair shake.

Constructivists have had a good hundred years to show us what they can do.

I say it's time to move on.

[Thank you, Chris Correa.]

NotTheWholeStory

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Carolyn's right that Everyday Math can't be blamed for the sorry state of college freshmen's ability to add fractions.

I haven't been able to track down the first printing, but EVERYDAY MATH seems to date back to around 1993 or thereabouts.

Garelick reports that approximately 10% of U.S. schools have now adopted E-Math, and I read just this week that another 10% of U.S. schools have adopted one of the other constructivist math curricula. (I've forgotten the source, or I'd link -- sorry.)

Of kids entering college this year, only a small percentage will have spent much time using the latest crop of constructivist mathematics programs.

Of course, that's leaving aside the fact that constructivism has been part of ed school philosophy for a century.

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