Shadow

The Education Trust just published a new report on math teachers who shouldn’t be teaching math, according to NCLB. The numbers do not look good.

“As a nation, we must commit ourselves to ensuring that all students – no matter where they live – are taught by strong teachers,” said Kati Haycock, president of The Education Trust. “It’s astonishing that in America, a country dedicated to opportunity for all, we are still assigning our most vulnerable children to the teachers with the weakest capacity to teach them what they need to know.”

Nonetheless, "These are bright spots in an otherwise bleak landscape," according to Haycock.

• The University of Texas at Austin, the University of NorthCarolina System, and the University System of Georgia are working to develop strong teachers to fill local needs, both for the projected number of new teachers overall as well as in subject-specific areas.
• Louisiana committed to overhauling all teacher-preparation programs in the state, both traditional and alternative routes.As part of this overhaul, the state examines student achievement data and holds teacher-preparation programs accountable for their graduates’ ability to improve student learning.
• Teacher residency programs in such places as Boston andChicago are modeled after the medical school formula. These place teacher candidates for one year in the school in which they will work, so they can learn alongside accomplished mentor teachers before being assigned to their own classrooms.
• Denver Public Schools and Guilford County (N.C.) Public Schools provide financial incentives to attract the best teachers to work in hard-to-staff subjects and schools.

A mixed list if you ask me. 

Like the Reading Panel back in 2000, this Math panel report is likely to have significant impace on math education in the US. The goals, though, are moderate:

Benchmarks in Math Education Fluency With Whole Numbers

1 By the end of Grade 3, students should be proficient with the addition and subtraction of whole numbers.

2 By the end of Grade 5, students should be proficient with multiplication and division of whole numbers.

Fluency With Fractions

1 By the end of Grade 4, students should be able to identify and represent fractions anddecimals, and compare them on a number line or with other common representations offractions and decimals.

2 By the end of Grade 5, students should be proficient with comparing fractions and decimalsand common percents, and with the addition and subtraction of fractions and decimals.

 

More info: National Mathematics Advisory Panel

National Mathematics Advisory Panel Releases Final Report

On March 13, 2008, the National Mathematics Advisory Panel presented its Final Report to the President of the United States and the Secretary of Education. Copies of these ground-breaking reports, rich with information for parents, teachers, policy makers, the research community, and others, are provided below.

Foundations for Success: Report of the National Mathematics Advisory Panel

Final Report PDF (851 KB) | Word (1 MB)

Draft Task Group Reports

• Conceptual Knowledge and Skills Word (1.3 MB)
• Learning Processes Word (7.9 MB)
• Instructional Practices Word (2.9 MB)
• Teachers Word (1.2 MB)
• Assessment Word (876 KB)

Draft Subcommittee Reports

• Standards of Evidence PDF (68 KB) | Word (276 KB)
• Instructional Materials Word (958 KB)
• National Survey of Algebra Teachers for the National Math Panel PDF (4.1 MB) | Word (3.2 MB)

Fact Sheet

Panel members:

Conceptual Knowledge and Skills

• Francis "Skip" Fennell, Chair
• Larry Faulkner
• Liping Ma
• Wilfried Schmid
• Sandra Stotsky

Learning Processes

• David Geary, Chair
• Wade Boykin
• Susan Embretson
• Valerie Reyna
• Bob Siegler
• Dan Berch, Ex Officio

Instructional Practices

• Russell Gersten, Chair
• Camilla Benbow
• Douglas Clements
• Bert Fristedt
• Tom Loveless
• Vern Williams
• Joan Ferrini-Mundy, Ex Officio

More at http://www.ed.gov/about/bdscomm/list/mathpanel/bios/index.html#panel

 

Tried to explain 3-(-5)=8 to my daughter today to no avail. Where I went wrong was to try to explain the whole thing on a number line, but I confused myself before confusing Jessie — I guess she was not bothered by the math challenge at all; she just watched me fumbling for an explanantion and laughed to her tears. She finally agreed  3-(-5)=3-(0-5)=3-0+5=8, but I know she was just tired of me.

I had to turn to wikipedia for answer, and as always, it never fails me.  

Subtracting a negative is equivalent to adding the corresponding positive:

5 − (−2) = 5 + 2 = 7(if you have a net worth of $5 and you get rid of a debt of $2, then your new net worth is $7).

Also:

−8 − (−3) = −5(if you have a debt of $8 and get rid of a debt of $3, then you still have a debt of $5).

I admit Wikipedia’s examples make more sense than what I tried to do, but I still think the number line will work. It gotta work. It’s the mother of all numerical representations! (Or is it?)

While on Wikipedia, I found that I am not the first — or last — person to puzzle over the concept of negativity. 

Negative numbers were not well-understood until modern times. As recently as the 18th century, the Swiss mathematician Leonhard Euler believed that negative numbers were greater than infinity, and it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless.⁴

Wikipedia lead me to another really interesting math site for middleschool students, where they have several Q&A sections on negative numbers. I find this discussion fascinatingly unsettling:

Math Forum: Ask Dr. Math FAQ: Negative Times a Negative

A Mathematical Explanation

If we can agree that a negative number is just a positive number multiplied by -1, then we can always write the product of two negative numbers this way:

   (-a)(-b) = (-1)(a)(-1)(b) = (-1)(-1)ab 

For example,

    -2 * -3 = (-1)(2)(-1)(3)
	
            = (-1)(-1)(2)(3)
	
            = (-1)(-1) * 6

So the real question is,

   (-1)(-1) = ?

and the answer is that the following convention has been adopted:

   (-1)(-1) = +1

This convention has been adopted for the simple reason that any other convention would cause something to break.

For example, if we adopted the convention that (-1)(-1) = -1, the distributive property of multiplication wouldn’t work for negative numbers:

   (-1)(1 + -1) = (-1)(1) + (-1)(-1)
	
        (-1)(0) = -1 + -1
	
              0 = -2

As Sherlock Holmes observed, "When you have excluded the impossible, whatever remains, however improbable, must be the truth."

Since everything except +1 can be excluded as impossible, it follows that, however improbable it seems, (-1)(-1) = +1.

It’s inadequate in my mind, not because of the lack of mathematical rigor, but because it basically says "it has to be the case" without giving me any intuition. Fascinating, because often times this is the level at which we learn and operate symbol systems — "this is how it is; take it or leave it". The reasons, the design principles are beyond most of us, and we rarely bother to ask. Eventually, Jessie will grow up like me, knowing (-1)*(-1)=1 like second nature but never asking why. Only the learners and (some) teachers, and maybe a few mathematicians, think deeper questions.

Such as this boy, who asked for a proof that 1 + 1 = 1. And he got it here on the Dr. Math forum.

Chris Correa » Teachers’ Knowledge of Fractions

The Journal of Contemporary Educational Psychology is publishing an article entitled Knowing and teaching fractions: A cross-cultural study of American and Chinese mathematics teachers. 90 U.S. teachers and 70 Chinese teachers participated in an assesment of pedagogical content knowledge.

First, the researchers compared U.S. and Chinese teachers’ knowledge of fractions with a 13-item test of the teachers’ understanding of fractions. The U.S. teachers did not fare well:

Don’t have access to this article from home, but I wonder how familiar the Chinese and American teachers were with the kind of questions and the ways questions are asked.

Chinese teachers spend a lot of time studying test problems. My teachers used to collect exams from other school districts, discuss solutions, and bet on the types of questions that would be on the next district test. Anybody who’s gone through SAT prep schools should know what it’s like. With the increasing pressure of high stakes testing, American teachers will eventually become test experts. Of course it will take another step to explain problem solving tricks to students.

I don’t dispute the benefits of being exposed to varieties of problems and being flexible in thinking, but doesn’t the paper suggest that Chinese teachers are better test takers/makers than American counterparts? Or maybe I am too cynical.

Chris Correa’s post today coincides with our discussion on the No Child Left Behind in Psy145. Although Chris focuses on math and our primary interest is reading, the simple-mindedness of the current trend of evidence-based policy-making is felt everywhere in education research.

Quoting Catherine Lewis, Rebecca Perry, and Aki Murata, Chris says:

The authors note that “faddism” is a problem in educational research. New ideas are proposed and evaluated all the time before we really understand the older ideas.

The root cause of educational faddism, in the view of some policymakers, is adoption of educational practices that have not been tested through randomized controlled trials. In contrast, we suggest that summative trials of lesson study — given how little is currently known about its nature and mechanisms — might actually contribute to making it a fad.

The danger is that researchers and policy-makers could dismiss a useful idea based on a premature randomized trial or two.

They conclude with six recommendations that center around their idea of a “local proof” model of how research can improve schooling. They also emphasize the value of refining educational practices througn research. The ideas are useful, but in many ways they are swimming against the current.

I’d be interested in reading about how Local Proof works. 

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…The stunning revelation for Greek philosophers such as Pythagoras, Plato and Aristotle, was that despite its chaotic appearance, the world was really a lawful place. What excited them was the thought that through the exercise of their intellects, they might hope to penetrate the world´s deepest secrets.

Before the flowering of Greek philosophy, life was shrouded in dark mystery. Humans might make laws to rule their cities but nature seemed lawless. Bad weather, famine and disease could strike at any time. It was impossible to be certain what fate had in store. All people could do was sacrifice to the gods and pray they would be treated kindly. Then the Greeks discovered mathematics. In the clarity of geometry and logic, philosophers found a gorgeous and reassuring certainty. It seemed that man at last had pierced the confused surface of reality to see the clockwork precision that lay beneath. The discovery of mathematics had an immense impact. It set in train an approach to knowledge that was to dominate the next 2,000 years. Where before people had relied on folk- wisdom and legend, Greek philosophers created the belief that the world could be understood through reason alone. In the symmetry of the circle and the elegant lines of the rectangle, there was the promise of perfect knowledge.