In this five option multiple-choice question, 22% of American eighth graders got the correct answer. We could analyze in purely statistical terms how much better than wild guessing this is, but it's clearly not much. In other words, the results essentially meant that the entire nation's soon-to-enter high school students did not understand the difference between a million and a billion.

## 2012-04-30

### How many is a billion?

In 1992, the National Assessment of Educational Progress, otherwise known as ``The Nation's Report Card'', presented the following question:

In this five option multiple-choice question, 22% of American eighth graders got the correct answer. We could analyze in purely statistical terms how much better than wild guessing this is, but it's clearly not much. In other words, the results essentially meant that the entire nation's soon-to-enter high school students did not understand the difference between a million and a billion.

In this five option multiple-choice question, 22% of American eighth graders got the correct answer. We could analyze in purely statistical terms how much better than wild guessing this is, but it's clearly not much. In other words, the results essentially meant that the entire nation's soon-to-enter high school students did not understand the difference between a million and a billion.

## 2012-04-26

### Decomposing numbers in kindergarten

Common Core standard K.OA.3 states, ``Decompose numbers less than or equal to
10 into pairs in more than one way, e.g., by using objects or drawings,
and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3
and 5 = 4 + 1).''

Although it falls under the general heading of addition and subtraction, the mathematical concept that this exercise is supposed to teach kindergartners is mystifying. By ``pairs'', does it really mean ``groups''? Is it an introduction to odd and even numbers? The commutative property? Transitivity? Algebra? Combinatorics? Or to impress on the kindergartner that both 4 + 1 and 2 + 3 equal 5?

Although it falls under the general heading of addition and subtraction, the mathematical concept that this exercise is supposed to teach kindergartners is mystifying. By ``pairs'', does it really mean ``groups''? Is it an introduction to odd and even numbers? The commutative property? Transitivity? Algebra? Combinatorics? Or to impress on the kindergartner that both 4 + 1 and 2 + 3 equal 5?

## 2012-04-24

### Addition and subtraction ad nauseam

The pre-CCSSI 2008

__Final Report of the National Mathematics Advisory Panel__, commissioned by the U.S. Department of Education, in a section entitled ``A Need for Coherence'', was bluntly critical of``...U.S. curricula [that] generally review and extend at successive grade levels many (if not most) topics already presented at earlier grade levels, while the top-performing countries are more likely to expect closure after exposure, development, and refinement of a particular topic. These critical differences distinguish a spiral curriculum (common in many subjects in U.S. curricula) from one built on developing proficiency—a curriculum that expects proficiency in the topics that are presented before more complex or difficult topics are introduced.''Every math teacher knows the concept of spiraling, the revisiting of old problems to ``reinforce'' concepts lest students should forget. The NMAP in no uncertain terms clearly rejected this approach, but what does Common Core do? It introduces addition and subtraction in kindergarten, ``students should see addition and subtraction equations,'' yet CCSSI is still covering addition and subtraction of whole numbers into the fourth grade:

## 2012-04-23

### Not a diatribe

This is not a blog written by those who oppose a national curriculum on the basis of an anti-government ideology. A common standard makes sense. The highest performing countries have them. The problem arises when you commit an entire nation to a substandard, untested curriculum. This blog reflects our singular commitment to a better mathematics education for all.

### The concept of none

Common Core's first major blunder is its treatment of nothing.

From CCSSI K.CC.3: ``Write numbers from 0 to 20. Represent a number of objects with a

written numeral 0-20 (with

Long before children read and write, they speak, and long before they write numbers, they count and conceptualize. As young children, we don't count starting at zero, we start at one.

At what age can a child abstract that when you take away everything, you have nothing and that is represented by the number zero? Obviously, it's a concept you introduce in stages: everyone knows ``none'' or ``nothing'' before you abstract to the number 0. We don't claim to know what the appropriate age is, but we KNOW this doesn't belong in kindergarten, before counting is mastered. We'd guess that by the end of first grade, after learning about ``taking away'' and subtraction, every child should understand both the concept and perhaps the number. But the steps have to be clearly set out as part of the curriculum; otherwise, ``0'' will likely be glossed over by teachers in the classroom as something trivial and obvious, which it is not.

CCSSI's ``Mathematical practice No. 2'' is: ``Reason abstractly and quantitatively.'' The concept of none is the first major abstract idea in mathematics, and it needs careful treatment.

[CCSSI does in fact mention the sequence from concept of none to digit 0...in the Mathematical Standards for High School, p. 58. Which elementary teachers are reading the high school standards?]

From CCSSI K.CC.3: ``Write numbers from 0 to 20. Represent a number of objects with a

written numeral 0-20 (with

**0 representing a count of no objects**).''Long before children read and write, they speak, and long before they write numbers, they count and conceptualize. As young children, we don't count starting at zero, we start at one.

At what age can a child abstract that when you take away everything, you have nothing and that is represented by the number zero? Obviously, it's a concept you introduce in stages: everyone knows ``none'' or ``nothing'' before you abstract to the number 0. We don't claim to know what the appropriate age is, but we KNOW this doesn't belong in kindergarten, before counting is mastered. We'd guess that by the end of first grade, after learning about ``taking away'' and subtraction, every child should understand both the concept and perhaps the number. But the steps have to be clearly set out as part of the curriculum; otherwise, ``0'' will likely be glossed over by teachers in the classroom as something trivial and obvious, which it is not.

CCSSI's ``Mathematical practice No. 2'' is: ``Reason abstractly and quantitatively.'' The concept of none is the first major abstract idea in mathematics, and it needs careful treatment.

[CCSSI does in fact mention the sequence from concept of none to digit 0...in the Mathematical Standards for High School, p. 58. Which elementary teachers are reading the high school standards?]

### Does a kindergartner need to know what a hexagon is?

From CCSSI: Kindergartners ``identify, name, and describe basic two-dimensional shapes, such as squares, triangles, circles, rectangles, and hexagons...''

Can every kindergartner visually distinguish between a square and a rectangle? Likely. Two dimensions and three dimensions? Dubious. And know what a hexagon is? No, it's age inappropriate. What's the point of pushing a kindergartner to know what a hexagon is? Is this going to lay the foundation to be college ready?

From CCSSI, a question that a teacher should ask a kindergartner and they are expected to understand and complete the task: “Can you join these two triangles with full sides touching to make a rectangle?”

``with full sides touching''? It sounds awkward because it is awkward.

Shouldn't a kindergartner be playing with blocks instead? Any kindergartner playing with blocks knows there are rectangular blocks and triangular blocks. If you put two triangular blocks one on top of the other, the top block will slide off. Why make this a goal-oriented and evaluated task?

Leave it to CCSSI to take the fun out of blocks.

Can every kindergartner visually distinguish between a square and a rectangle? Likely. Two dimensions and three dimensions? Dubious. And know what a hexagon is? No, it's age inappropriate. What's the point of pushing a kindergartner to know what a hexagon is? Is this going to lay the foundation to be college ready?

From CCSSI, a question that a teacher should ask a kindergartner and they are expected to understand and complete the task: “Can you join these two triangles with full sides touching to make a rectangle?”

``with full sides touching''? It sounds awkward because it is awkward.

Shouldn't a kindergartner be playing with blocks instead? Any kindergartner playing with blocks knows there are rectangular blocks and triangular blocks. If you put two triangular blocks one on top of the other, the top block will slide off. Why make this a goal-oriented and evaluated task?

Leave it to CCSSI to take the fun out of blocks.

## 2012-04-20

### Common Core's lack of humility

If nothing else, the statements in and around the standards are self-serving and self-aggrandizing. Why is this necessary? Let the standards speak for themselves.

To wit:

``The standards...are clear, understandable and consistent.''

``To deliver on the promise of common standards, the standards must address the problem of a curriculum that is 'a mile wide and an inch deep.' These Standards are

``What is important to keep in mind is that the progression in the Common Core State Standards is mathematically coherent and leads to college and career readiness at an internationally competitive level.''

Oh, really? No one knows what the outcome of CCSSI will be. Fast forward 12 years for a reality check when the first students graduate high school who had a lifetime of Common Core, and the people who made these highfalutin claims are already collecting their pensions.

To wit:

``The standards...are clear, understandable and consistent.''

``To deliver on the promise of common standards, the standards must address the problem of a curriculum that is 'a mile wide and an inch deep.' These Standards are

**.'' (emphasis added)***a substantial answer to that challenge*``What is important to keep in mind is that the progression in the Common Core State Standards is mathematically coherent and leads to college and career readiness at an internationally competitive level.''

Oh, really? No one knows what the outcome of CCSSI will be. Fast forward 12 years for a reality check when the first students graduate high school who had a lifetime of Common Core, and the people who made these highfalutin claims are already collecting their pensions.

### CCSSI coming soon to a school near you

The Common Core State Standards Initiative has been adopted and is being implemented by over 40 states and DC. This blog is taking a closer look at the math standards, and math teaching in general. A list of blog entries is at the right.

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