kitchen table math, the sequel: 4/25/10 - 5/2/10

Saturday, May 1, 2010

Did Congress Authorize Race to the Top?

Brookings Institute article.

There is nothing in the text of the ARRA, or in the portions of the two other statutes to which it points (the Elementary and Secondary Education Act and the America Competes Act), that authorizes, requires, or even suggests that states competing for funds would need to adopt common state standards, create more charter schools, evaluate teachers and principals based on gains in student achievement, emphasize the preparation of students for careers in science, technology, engineering, and mathematics, or restructure the lowest 5 percent of their schools.


Let's Do The Numbers
Economic Policy Institute Briefing Paper #263
(p. 7) In sum, Massachusetts’ willingness to permit the public to comment on its academic standards, combined with a few quirks in the weighting system, cost the state hundreds of millions of dollars.

Thursday, April 29, 2010

Are standards requirements? Are curricula products?

The push for standards is everywhere. But what are standards?

In engineering disciplines, engineers build products based on *requirements*.

Requirements are specifications of a problem to solve in order to build a product, or system or process. Requirements are NOT a design--they are not a solution. They specify key elements that a solution must have, should have, and would like to have, but properly written, they leave open what the actual solution is.

Examples of requirements are:
device must weigh less than 2.2 ounces;
device must operate properly during and afterw a fall from 8 inches into a bath of 110 degree liquid for not less than 6 seconds duration
device user input method must support touch typing at not less than 20 wpm

system performance requirements are things like:
system must detect intrusion within N seconds with probability of detection >=80% in noisy/cluttered environment

system must locate target within accuracy of 3 sq feet at range of 1.5 miles with false positive rate of no more than 1%

Excellent requirements are difficult to write. The best ones are (at least) unambiguous, concise, testable and/or measurable, and finite. Unambiguous requirements leave little room for confusion (and seldom have compound requirements in them.) Measurable requirements give you a way to tell performance; testable ones allow you a method to demonstrate you've met the requirement.

Requirements define inputs and outputs as well as what happens in between.

Given good requirements, a good design will meet them, and even more, will trace how they are met (if possible, every design choice points to a set of requirements, and on design choice exists without such requirements backing it up.) Every choice at every stage would be traceable to the requirements.

Among the stakeholders looking for a product/process/system, most aren't skilled at requirements writing. Explicit requirements are few and far between, and most must be teased out, so that derived requirements (that do meet the above criteria) can be created.

Given that, what in the world are standards in education? Are they supposed to be requirements? Of WHAT system/process/product? A classroom's learning? A classroom's teaching? Are they requirements from which one defines curricula? Is the curricula the product that meets the standards? Are students the product that are supposed to meet standards?

Reading the Common Core standards, they don't come close to meeting the criteria of requirements.
Here's an example:

1. Understand that multiplication of whole numbers is repeated addition. For example, 5  7 means 7 added to itself 5 times. Products can be represented by rectangular arrays, with one factor the number of rows and the other the number of columns.
2. Understand the properties of multiplication.
a. Multiplication is commutative. For example, the total number in 3 groups with 6 things each is the same as the total number in 6 groups with 3 things each, that is, 3  6 = 6  3.
b. Multiplication is associative. For example, 4  3  2 can be calculated by first calculating 4  3 = 12 then calculating 12  2 = 24, or by first calculating 3  2 = 6 then calculating 4  6 = 24.
c. 1 is the multiplicative identity.
d. Multiplication distributes over addition (the distributive property). For example, 5  (3 + 4) = (5  3) + (5  4).

So, apparently, these are *student* requirements.
They mean to be saying
"THE STUDENT SHALL understand that"

But what student? All students? A typical student? A student who passed the prior year's standards?

What does it mean to understand that 1 is the multiplicative identity? Does it mean to be able to recite that phrase? use that fact in a problem? use that fact in N problems, and get the answer right M times out of N problems? does it mean being able to cite the use of the multiplicative identity in order to solve the problem?

These standards aren't finite, concise, unambiguous, and without guidance, certainly aren't testable.

Where does that leave curricula? Since the standards are "THE STUDENT SHALL" rather than "the system shall", how can any curricula meet the requirements of what a student is required to learn?

If curricula aren't solutions to the problem defined by the standards requirements, then what are curricula? And what are teachers or students or schools? Stakeholders ? Or something else?

Wednesday, April 28, 2010

Why do we insist on teaching fractions like we teach poetry?

...asks professor Hung-Hsi Wu of Berkeley at last week's NCTM (National Council of Teachers of Mathematics) Conference. Once you learn algebra, precision is most desirable, mathematics can not be taught in an ambiguous manner. He brings Mark Twain, Shakespeare and Keats into an informative and, for me, entertaining conference session.  (At one point, he turned the definition of multiplication of fractions into free form verse!) From his presentation:
Consider Hamlet's comment on Denmark after his father's death:
'Tis an unweeded garden
That grows to seed; things rank and gross in nature
Possess it merely.
Compare it with the definition of 3/4 :
Take a pizza (or a fraction bar) and divided it into 4 equal parts. Take 3 parts.
My favorite quote from his presentation came after working to understand the product formula (is there an official name for this?): a/b x c/d = ac/bd. He said something along the lines of:

"You've worked so hard to understand, don't you need a treat? Here's the standard algorithm. You can almost use it mindlessly."

At which point people in the audience started snickering & guffawing. Wu continued:

"We drive mindlessly. Do we consider the internal combustion engine when we do so?"

He insists that computation is a part of mathematics:
Analogies and metaphors have a place in mathematics. They can be very helpful in the understanding of precise concepts and reasoning. However, it is a mistake to allow them to replace precise concepts and reasoning.
Let us hope that fractions will be taught with less poetry, but with more emphasis on
precise definitions, and
precise reasoning
Want to enjoy the whole presentation in your own home? Point your browser to:

Karen Pryor on paying students to learn

Why economists can't learn something about reinforcement theory is beyond me.