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.
Moreover, the professional test makers who devised and vetted this question may not understand mathematics significantly better than the students who took it. In many parts of the world today, a billion means—and in the UK and Commonwealth countries until the mid 1970's, a billion meant—10¹², not 10⁹, as it does in the US. For that reason, a 10¹² billion is sometimes referred to as ``a million million'' to avoid confusion. On the NAEP question, 40% of students chose answer A, but we don't know who chose that answer because they guessed, or thought it was right because of some faulty logic or calculation, or actually got the ``correct'' answer because they immigrated from or learned from immigrant parents of a 10¹² nation.
Regardless, we can read the news that Town A has a budget deficit of $1 million and City B has a budget deficit of $1 billion and be confident that a large segment of the American population does not correctly understand the scope of the magnitude of the difference between these two numbers.
Curriculum specialists have had 20 years to ruminate on the million/billion problem, which brings us to a discussion of place value.
In the following 1992 NAEP question, 89% of eighth graders answered correctly:
It should be no surprise that this question would be solved correctly by a vast majority of students, because students basically ``get'' place value and number names when it comes to whole numbers less than a million. It's not a difficult concept and it's not generally cited by educators as one of their students' central stumbling blocks. Even without calculators, elementary school students should and do learn the mechanics of adding, subtracting and multiplying 2, 3 and 4 digit numbers and naming and understanding the results.
CCSSI, though, manifests unnecessary worry about a non-issue. It devotes outsized attention to the basic concept of place value, and pushes the start all the way back to kindergarten. Here are excerpts:
Kindergarten:
``Work with numbers 11–19 to gain foundations for place value.''
First grade:
``Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:
a. 10 can be thought of as a bundle of ten ones — called a “ten.”
b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).''
and alsoa. 10 can be thought of as a bundle of ten ones — called a “ten.”
b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).''
``4. Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.
5. Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.
6. Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.''
Second grade:
``1. Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases: a. 100 can be thought of as a bundle of ten tens — called a “hundred.”
b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
2. Count within 1000; skip-count by 5s, 10s, and 100s.
3. Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
4. Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.
Use place value understanding and properties of operations to add and subtract.
5. Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
6. Add up to four two-digit numbers using strategies based on place value and properties of operations.
7. Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
8. Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.
9. Explain why addition and subtraction strategies work, using place value and the properties of operations.''
Third grade:
Use place value understanding and properties of operations to perform multi-digit arithmetic.
1. Use place value understanding to round whole numbers to the nearest 10 or 100.
2. Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
3. Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.
Fourth grade:
Generalize place value understanding for multi-digit whole numbers.
1. Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.
2. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
3. Use place value understanding to round multi-digit whole numbers to any place.
(Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.)1. Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.
2. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
3. Use place value understanding to round multi-digit whole numbers to any place.
Fifth grade:
1. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
We won't pick apart these standards line by line, but whereas some of it is obviously necessary, some is misplaced, and many of its emphases could be eliminated with no loss of structure, continuity or completeness. Some of it is downright bizarre, such as ``Explain why addition and subtraction strategies work, using place value and the properties of operations.'' We've highlighted in red most of what should be left in, but not necessarily at the grade level CCSSI specifies.
Here is a summary of how the treatment of place value of whole numbers would be better organized:
¶Kindergartners do not need to be exposed to place value. It is enough to count and measure, and they don't need to understand why ``10'' looks so much different than ``9''.
¶First graders can skip a formal exposure to place value as well, but they should be introduced to the concept of groups of 10, including problems of addition and subtraction. This should be done with pictures and manipulatives, such as a boxes of 10 balls and loose balls, not with written numbers. First graders should add and subtract, but by counting only. ``Composing a `ten'" is too abstract, too early. Boxes of balls teach the concept; that's enough. They should end the year with the mandate to memorize the addition tables during the summer, which they are going to need to apply rapidly in the 2nd grade.
¶Second graders should be exposed to bundles of sticks and loose sticks, with lots of examples, as a formal introduction to place value. A tied bundle of 10 sticks is a ``ten''. Then the concept will be fresh in students' minds and lead naturally to the abstraction of carrying and borrowing. But why stop at 10? A bundle of 10 tens is a hundred. There's no reason that tens and hundreds cannot be introduced in close proximity in the same year; in fact, they belong together so students can recognize the critically important patterns of place value. (Some students will extend the notion to understand that you could bundle 10 hundreds, etc.) CCSSI gets this totally wrong by separating tens and hundreds into different years. The capacity of second graders is far higher than kindergartners and they are ready to add and subtract to hundreds.
¶Third graders should be learning about thousands and tens and hundreds of thousands, but the notation of grouping places into sets of 3 with commas, such as 1,000 and 100,000 is not yet necessary. (Reason for thousands: they will obtain numbers in the thousands when they are multiplying 2 and 3 digit numbers, which belongs in the third grade curriculum.) Once students understand the concepts that (1) moving one digit to the left means a tenfold increase, and (2) moving one digit to the right is a tenfold decrease (described in terms of division, because third grade is the year to learn division as well, but NOT described in terms of 1/10–fractions come later) then moving places to the right will lead naturally into the topic of decimals later on. (A formal learning of decimals should precede a formal learning of fractions, but more on that in another post.)
¶Around fourth grade, students are ready to learn about really big numbers. Why is it important at this age? One, because elementary students are obviously capable of the material after the proper foundation has been laid, but also because in their science classes, they may be studying about dinosaurs or planets, and how can you fathom the ages of dinosaurs or the distances of planets without understanding really big numbers? Also, students should be exposed to (and finish the topic of) whole numbers before a formal introduction to decimals and fractions. The CCSSI stops at 1,000,000. Why? When are the names and writing notation of big numbers supposed to be learned? A clear standard is missing. There's no need to impose such artificial limits, and we can only lament that perhaps the supporters of CCSSI and other modern curricula have learned nothing from the 1992 NAEP million/billion debacle.
When it comes to place value, CCSSI begins too early and ramps up too slowly. This perpetuates the problem of spiraling, repeating and reviewing old material, rather than following the preferred route of (1) understanding the concept, (2) a formal introduction and abstraction, and (3) mastery and closure.
